0

Poisson distribution - class-XII

Description: poisson distribution
Number of Questions: 91
Created by:
Tags: maths probability distributions probability distribution business maths random variables and probability distribution
Attempted 0/91 Correct 0 Score 0

In a poisson distribution, the variance is $m$ . The sum of terms in odd places in the distribution is 

  1. $e^{-m}$

  2. $e^{-m} \cos \, h \, (m)$

  3. $e^{-m} \sin \, h \, (m)$

  4. $e^{-m} \cot \, h \, (m)$


Correct Option: B
Explanation:

$e^{-m}\cos h(m)$


 lf the mean is $\lambda$ and the variance is $\sigma^{2}$ in a Poisson distribution, then

  1. $\displaystyle \lambda=\frac{1}{2}\sigma^{2}$

  2. $\displaystyle \sigma^{2}=\frac{1}{2}\lambda$

  3. $\lambda=\sigma^{2}$

  4. $\sigma^{2}=\lambda^{2}$


Correct Option: C
Explanation:

For a Poisson distribution, mean and variance are equal .
i.e.$\lambda = \sigma^2$

If the mean of P.D. is 5, then the variance of the same distribution is

  1. $25$

  2. $10$

  3. $5$

  4. $15$


Correct Option: C
Explanation:

The mean of P.D. is $5$. 
In P.D,  the variance and mean of the same distribution is same.
So, variance = $5$

If ${\overline{x}}$ and $\sigma^{2}$ are mean and variance of poisson distribution, then

  1. $\overline{x}>\sigma^{2}$

  2. $\overline{x}<\sigma^{2}$

  3. $\overline{x}=\sigma^{2}$

  4. $\overline{x}+\sigma^{2}=1$


Correct Option: C
Explanation:

For a Poisson distribution, mean ($\mu$) and variance ($\sigma^2$ )are equal .
i.e.$\mu = \sigma^2$

The S.D. of poisson distribuition whose mean is $\lambda $ is

  1. $\lambda$

  2. $\sqrt{\lambda}$

  3. $\lambda^{2}$

  4. $\displaystyle \frac{1}{\sqrt{\lambda}}$


Correct Option: B
Explanation:

Mean = $ \lambda$
In Poisson distribution, the expected value of a Poisson-distributed random variable is equal to that is mean and is equal to its variance. 
Variance =  $ \lambda$
S.D = $ \sqrt {variance} =\sqrt { \lambda  } $

If the mean of Poisson distribution is $\displaystyle \frac{1}{2}$, then the ratio of $P(X=3)$ to $P(X=2)$ is 

  1. 1:2

  2. 1:4

  3. 1:6

  4. 1:8


Correct Option: C
Explanation:

$P(X=3)/P(X=2)$
$=\dfrac{\lambda^{3}(2!)}{3!(\lambda^{2})}$
$=\dfrac{\lambda}{3}$
$=\dfrac{1}{2(3)}$
$=\dfrac{1}{6}$
The ratio is $1:6$

In a poisson distribution, the probability of $0$ success is $10$%. The mean of the distribution is equal to

  1. $\log _{10}e$

  2. $\log _{e}10$

  3. $0$

  4. $\dfrac{1}{10}$


Correct Option: B
Explanation:

Given $P(X=0)=.1$
$\Rightarrow e^{-\lambda}=\cfrac{1}{10}\therefore \lambda = \log _e 10$

The parameter $\lambda $ of poisson distribution is always

  1. zero

  2. 1

  3. -1

  4. a finite positive value


Correct Option: D
Explanation:

In Poisson distribution the  parameter $(  > 0)$, it is always greater than zero or a finite positive value.

If X is a poisson variable with parameter 0.09,then its S.D. is

  1. 0.009

  2. 0.3

  3. 0.03

  4. 0.09


Correct Option: B
Explanation:

X is a Poisson variable with parameter 0.09.
In Poisson distribution,  
Poisson variable(X) is equal to  expected value of X and also to its variance. 
 S.D =  
$\sqrt {variance}  =  \sqrt { X } =\sqrt { 0.09 } =0.3$

The standard deviation of P.D. is 1.5, then its mean is

  1. 1.5

  2. 2

  3. 2.25

  4. 3.25


Correct Option: C
Explanation:

Given $\sigma =1.5$
We know that for a Poisson's distribution,mean and variance are equal.
$\mu=\sigma^2$
$\Rightarrow \mu =2.25$

If the mean of poisson distribution is $16$, then its S.D. is

  1. $16$

  2. $4$

  3. $10$

  4. $15$


Correct Option: B
Explanation:

$S.D= \sqrt { \lambda  }$

Mean$= \lambda$
Therefore S.D.$= \sqrt { 16   } \=4 $

Six coins are tossed $6400$ times. The probability of getting $6$ heads $x$ times using poison distribution is

  1. $6400{e^{ - x}}$

  2. $\frac{{6400{e^{ - x}}}}{{x!}}$

  3. $\frac{{{e^{ - 100}}{{100}^x}}}{{x!}}$

  4. ${e^{ - 100}}$


Correct Option: C
Explanation:

Therefore, the probability of getting $6$ heads with $6$ coins $ = {\left( {\frac{1}{2}} \right)^6} = \frac{1}{{64}} = P\left( { > {a _y}} \right)$
Then, 
$\eta P = 6400 \times \frac{1}{{64}} = 100 = m\left( {{a _y}} \right)$
So, by poison's law, $P\left( {x = n} \right) = \frac{{{e^{ - m}}{m^x}}}{{x!}}$
$ = \frac{{{e^{ - 100}}{{100}^x}}}{{x!}}$

A examinations consists of $8$ questions in each of which  one of the $5$ alternatives is the correct one. On the assumption that a candidate who has done no preparatory  work, chooses for each questions any one of the five alternatives with equal probability, then the probability that he gets more than one correct answer is equal to:

  1. ${\left( {0.8} \right)^8}$

  2. $3{\left( {0.8} \right)^8}$

  3. $1-{\left( {0.8} \right)^8}$

  4. $1-3{\left( {0.8} \right)^8}$


Correct Option: D
Explanation:
Probability of an answers to be correct $=\dfrac{1}{5}=0.2$

Probability of an answers not to be correct $=1-0.2$
$=0.8$

Probability $\left(more\ than\ 1\ correct\right)=1-P\left(0\ correct\right)-P\left(1\ correct\right)$

$=1-^{8}{C} _{0}{\left(0.8\right)}^{8}-^{8}{C} _{1}\left(0.2\right) \left(0.8\right)$

$=1-{\left(0.8\right)}^{8}-1.6{\left(0.8\right)}^{7}$

$=1-{\left(0.8\right)}^{8}-2{\left(0.8\right)}^{8}$

$=1-3{\left(0.8\right)}^{8}$

$D$ is coorect.

There are twenty bags each containing 10 bulbs and it is knows that no bag contains more than 5 defective bulbs and 3 bags have 5 defective bulbs. 4 bags have atleast 4 defective bulbs, 5 bags have atleast 3 defective bulbs, 6 bags have atleast 2 defective bulbs and 7 bags have atleast 1 defective bulb. Then the ratio of total defective bulbs is to non-defective bulbs is

  1. $\dfrac { 4 }{ 7 } $

  2. $\dfrac { 3 }{ 7 } $

  3. $\dfrac { 2 }{ 7 } $

  4. $\dfrac { 1 }{ 7 } $


Correct Option: A

For a Poission distribution which pair has same value.

  1. (Mean, Std. Deviation)

  2. (Variance, Standard Deviation)

  3. (Mean, Variance)

  4. None of these


Correct Option: C
Explanation:
For Poission distribution we have $\displaystyle P\left ( r \right )=\dfrac{e^{-\pi }\lambda ^{r}}{r^{i}};\left ( r=0, 1, 2,...\infty  \right)$ 
$\displaystyle$ mean $\displaystyle =\dfrac{\sum f _{i}x _{i}}{\sum f _{i}}=\sum _{r=0}^{\infty }rP\left ( r \right ),\sum f _{i}P\left ( r \right )=1$$\displaystyle =0+\lambda e^{-\lambda }+2\dfrac{\lambda ^{2}e^{-\lambda }}{2!}+3\dfrac{\lambda ^{3}e^{-\lambda }}{3!}+.....\infty$ $\displaystyle =\lambda e^{-\lambda }\left ( 1+\dfrac{\lambda }{11}+\dfrac{\lambda ^{2}}{2!}+\dfrac{\lambda ^{3}}{3!}+.....\infty  \right )$$\displaystyle =\lambda e^{-\lambda }.e^{\lambda }=\lambda $
Similarly, $\displaystyle o^{2}=$ Variance $=\sum _{r=0}^{\infty }r^{2}P\left ( r \right )-\left ( \sum _{r=0}^{\infty } rP\left ( r \right ) \right )^{2}$ $=\displaystyle \lambda e^{\lambda }\left ( e^{-\lambda }+\lambda e^{-\lambda } \right )-\lambda ^{2}=\lambda =$ mean 
$\therefore$ mean$=$variance

For a poission distribution variable $X$ is such that $P(X = 2) = 9 P(X= 4) + 90 P(X= 6)$ the mean is

  1. $2$

  2. $3$

  3. $1$

  4. None of these


Correct Option: C
Explanation:
For $P.D. P(X=r)=\displaystyle \dfrac{e^{-\lambda}\lambda ^{r}}{r!},r=0,1,2,\cdots$
$\therefore \displaystyle \dfrac{e^{-\lambda}\lambda ^{2}}{2!}=\dfrac{9e^{-\lambda}\lambda ^{4}}{4!}+90 \dfrac{e^{-\lambda}\lambda ^{6}}{6!}$        (given)
$\Rightarrow \displaystyle\dfrac{\lambda ^{2}}{2}=\dfrac{9}{24}\lambda ^{4}+\dfrac{90}{720}\lambda ^{6}$ 
$\Rightarrow \lambda ^{4}+3\lambda ^{2}-4=0$
$\Rightarrow \left ( \lambda^{2}+4 \right )\left ( \lambda^{2}-1 \right )=0=>\lambda= \pm 1$
$\Rightarrow \lambda=1$ as $\lambda>0$ and $(\lambda^{2}+4=0$ impossible$)$
$\therefore$ mean$=\lambda=1$

For a Poission distribution, which of the following is true

  1. $Mean = Mode$

  2. $Median = S.D.$

  3. $Mean = Variance$

  4. $Median = Variance$


Correct Option: C
Explanation:
For Poission distribution we have $\displaystyle P\left ( r \right )=\dfrac{e^{-\pi }\lambda ^{r}}{r^{i}}\left ( r=0, 1, 2,...\infty  \right)$ 
$\displaystyle$ mean $\displaystyle =\dfrac{\sum f _{i}x _{i}}{\sum f _{i}}=\sum _{r=0}^{\infty }rP\left ( r \right ),\sum f _{i}P\left ( r\right )=1$$\displaystyle =0+\lambda e^{-\lambda }+2\dfrac{\lambda ^{2}e^{-\lambda }}{2!}+3\dfrac{\lambda ^{3}e^{-\lambda }}{3!}+.....\infty$ $\displaystyle =\lambda e^{-\lambda }\left ( 1+\dfrac{\lambda }{11}+\dfrac{\lambda ^{2}}{2!}+\dfrac{\lambda ^{3}}{3!}+.....\infty  \right )$$\displaystyle =\lambda e^{\lambda }.e^{\lambda }=\lambda $
similarly, $\displaystyle o^{2}=$ Variance $=\sum _{r=0}^{\infty }r^{2}P\left ( r \right )-\left ( \sum _{r=0}^{\infty } rP\left ( r \right ) \right )^{2}$ $=\displaystyle \lambda e^{\lambda }\left ( e^{-\lambda }+\lambda e^{-\lambda } \right )-\lambda ^{2}=\lambda =$ mean 
$\therefore \text{mean}=\text{variance}$

At a telephone enquiry system the number of phone calls regarding relevant enquiry follow Poisson distribution with a average of 5 phone calls during IO-minute time intervals. The probability that there is at the most one phone call during a 10-minute time period is

  1. $\displaystyle \frac{6}{5^{e}}$

  2. $\displaystyle \frac{5}{6}$

  3. $\displaystyle \frac{6}{55}$

  4. $\displaystyle \frac{6}{e^{5}}$


Correct Option: D
Explanation:

According to Poisson distribution
$\displaystyle P(X=r)=\frac{e^{-m}m^{r}}{r!}$
$\displaystyle \therefore P(X\leq 1)=P(X=0)+P(X=1)$
$\displaystyle =e^{-m}+\frac{e^{-m} m}{1!}$
Given m $\displaystyle =$mean$ = 5 $
$\displaystyle \therefore P(x \leq 1)=e^{-5}+5\times e^{-5}=e^{-5}(1+5)=\frac{6}{e^{5}}$

The probability of r successes in case of poissons distrbution is

  1. $\dfrac{e^{\gamma }m}{\angle \gamma }$

  2. $\dfrac{\gamma ^{m}e^{m}}{\angle \gamma }$

  3. $\dfrac{e^{m}\gamma }{\angle \gamma }$

  4. $\dfrac{e^{-m}m^{r}}{\angle \gamma }$


Correct Option: D
Explanation:

in Poisson distribution, probability $ P(x; m  )=  \dfrac { { e }^{ - m  }{ m }^{ x } }{ \angle x } $
$m$ = mean  ;   $x$= number of success     $ \angle x =$ factorial $ x $
 For $r$ success  $x= r$;        
  $ P(r; m )=  \dfrac { { e }^{ -m  }{ m  }^{ r } }{ \angle r } $ 

A random variable $X$ has Poisson distribution with mean $2$. Then $P(X > 1.5)$ equals

  1. $2/e^{2}$

  2. $0$

  3. $1-\dfrac{3}{e^{2}}$

  4. $\dfrac{3}{e^{2}}$


Correct Option: C
Explanation:

For Poisson distribution of mean = $\mu = 2 $ 
 P$(x ; \mu) = \frac { { e }^{ -\mu  }{ \mu  }^{ x } }{ x! }$
$P(X>1.5) = 1 - [P(X=0) + P(X=1)]$ 

P$(0; 2) = \dfrac { { e }^{-2}{2}^{0}}{0!}={e}^{-2}$
 P$(1; 2) = \dfrac { { e }^{-2}{2}^{} } {1!}=2{e}^{-2}$
$P(X>1.5) = 1 - [P(X=0) + P(X=1)] $
$ = 1 - [{e}^{-2}+ 2{e}^{-2} ] = 1 - 3  {e}^{-2} = 1 - \frac { 3 }{ { e }^{ 2 } } $

If $X$ is a random poisson variate such that $\alpha =p(X=1)=p(X=2)$, then $p(X=4)=$

  1. $2\alpha $

  2. $\dfrac{\alpha }{3}$

  3. $\alpha e^{-2}$

  4. $\alpha e^{2}$


Correct Option: B
Explanation:

In  Poisson distribution such that  $ \alpha = p(X=1)=p(X=2) $ 
        
$   p(x; \mu) = \dfrac { { e }^{-\mu}{\mu}^{x}}{x!} $  
       => $ \alpha = \dfrac { { e }^{-\mu}{\mu}^{1}}{1!} = \dfrac { { e }^{-\mu}{\mu}^{2}}{2!}  $       =>   $ \alpha =  { e }^{-\mu}{\mu}  $
                 =>  $ { e }^{-\mu}{\mu} = \dfrac { { e }^{-\mu}{\mu}^{2}}{2} $
                 =>   $ \mu = 2 $ 
  $   p(4; \mu) = \dfrac { { e }^{-\mu}{\mu}^{4}}{4!} =  \dfrac { { e }^{ -\mu  }{ \mu \times { \mu  }^{ 3 } } }{ 24 }  = \dfrac { \alpha {\times { 2 }^{ 3 } } }{ 24 }  = \dfrac { \alpha  }{ 3 } $  

The variance of P.D. with parameter $\lambda $ is

  1. $\lambda $

  2. $\sqrt{\lambda }$

  3. $\dfrac{1}{\lambda}$

  4. $\dfrac{1}{\sqrt {\lambda}}$


Correct Option: A
Explanation:

$E[{ X }^{ 2 }]= \sum _{ k=0 }^{ \infty  }{ k^{ 2 } } \sum _{  }^{  }{  } \dfrac { 1 }{ k! } \lambda ^{ k }e^{ -\lambda  }\ \ = ^{ k }e^{ -\lambda  }\sum _{ 1 }^{ \infty  }{ k\dfrac { 1 }{ (k1)! } \lambda ^{ k-1 } } \= ^{ k }e^{ -\lambda  }(\sum _{ 1 }^{ \infty  }{ (k-1)\dfrac { 1 }{ (k1)! } \lambda ^{ k-1 } } ) +\sum _{ 1 }^{ \infty  }{ \dfrac { 1 }{ (k1)! } \lambda ^{ k-1 } } )\ =^{ k }e^{ -\lambda  }(\sum _{ 2 }^{ \infty  }{ (\lambda \dfrac { 1 }{ (k2)! } \lambda ^{ k-2 } } )+ \sum _{ 1 }^{ \infty  }{ \dfrac { 1 }{ (k1)! } \lambda ^{ k-1 } } )\ = ^{ k }e^{ -\lambda  }(\sum _{ i=0 }^{ \infty  }{ (\lambda \dfrac { 1 }{ i! } \lambda ^{ i } } )+ \sum _{ j=0 }^{ \infty  }{ \dfrac { 1 }{ j! } \lambda ^{ j } } )\ = ^{ k }e^{ -\lambda  }(\lambda e^{ \lambda  }+e^{ \lambda  })\  \ = { \lambda  }^{ 2 }+ \lambda \ \ Variance= E[{ X }^{ 2 }]- { E[X] }^{ 2 }\ = { \lambda  }^{ 2 }+ \lambda - { \lambda  }^{ 2 }\ = \lambda \ \ \ $

If a random variable $X$ has a poisson distributionsuch that $P(X=1)=P(X=2)$, its mean and varianceare

  1. $1,1$

  2. $2, 2$

  3. $2, 3$

  4. $2,4$


Correct Option: B
Explanation:

In  Poisson distribution such that $P(X=1)=P(X=2)$
    
$ P(x; \mu) = \dfrac { { e }^{-\mu}{\mu}^{x}}{x!}$
          =>  $ P(1; \mu)  =  P(2; \mu) $
         =>  $ \dfrac { { e }^{-\mu}{\mu}^{1}}{1!} = \dfrac { { e }^{-\mu}{\mu}^{2}}{2!}  $ 
          => $ 1 =$ $ \dfrac {\mu}{2} $ 
         => $ \mu = 2 $
 In Poisson distribution Variance $(m)$ is equal to mean
       Mean = Variance  = $2 $

If m is the variance of P.D., then the ratio of sum of the terms in even places to the sum of the terms in odd places is

  1. $e^{-m}\cosh m$

  2. $e^{-m}\sinh m$

  3. $\coth m$

  4. $\tanh m$


Correct Option: D
Explanation:

Sum of the terms in Even places will be 
$=e^{-m}[m+\dfrac{m^{3}}{3!}+\dfrac{m^{5}}{5!}+...]$

$=e^{-m}\dfrac{1}{2}[e^{m}-e^{-m}]$ ...(i)
Sum of the terms in Odd places will be 
$=e^{-m}[1+\dfrac{m^{2}}{2!}+\dfrac{m^{4}}{4!}+...]$

$=e^{-m}\dfrac{1}{2}[e^{m}+e^{-m}]$ ...(ii)
Taking the ratio of i and ii, we get 
$\dfrac{i}{ii}$

$=\dfrac{e^{-m}\dfrac{1}{2}[e^{m}-e^{-m}]}{e^{-m}\dfrac{1}{2}[e^{m}+e^{-m}]}$

$=\dfrac{e^{m}-e^{-m}}{e^{m}+e^{-m}}$

$=\dfrac{2sinhm}{2coshm}$

$=tanh(m)$.


If ${m}$ is the variance of Poisson distribution, then sum of the terms in even places is

  1. $e^{-m}$

  2. $e^{-m}\cosh m$

  3. $e^{-m}\sinh m$

  4. $e^{-m}\coth m$


Correct Option: C
Explanation:

If m is the variance of P. D, then  P$(x; \mu) = \dfrac { { e }^{-\mu}{\mu}^{x}}{x!} $
Sum of the terms in even places = [ P$(1; \mu) + P(3; \mu)  + P(5; \mu) + .........$ ]


                                       =  [$\dfrac { { e }^{-\mu}{\mu}^{1}}{1!} + \dfrac { { e }^{-\mu}{\mu}^{3}}{3!} + \dfrac { { e }^{-\mu}{\mu}^{5}}{5!} + ....... $]

                                      =  $  { e }^{-\mu} [ \mu + \dfrac {{\mu}^{3}}{3!} + \dfrac {{\mu}^{5}}{5!} + .......]$       --------------------- (1)

  Since   ${ e }^{ x }=1+\dfrac { x }{ 1! } +\dfrac { { x }^{ 2 } }{ 2! } +\dfrac { { x }^{ 3 } }{ 3! } +\dfrac { { x }^{4} }{ 4! } +....$
    ${ e }^{ -x }=1+\dfrac {( -x )}{ 1! } +\dfrac { { (-x) }^{ 2 } }{ 2! } +\dfrac { {( -x) }^{ 3 } }{ 3! } +\dfrac { { (-x) }^{4} }{ 4! } +....$ 

on subtracting , we get  
 ${e}^{x} - { e }^{-x} =  2[ x + \dfrac {{x}^{3}}{3!} + \dfrac {{x}^{5}}{5!} + .....] $
$ \dfrac {{e}^{x} - { e }^{-x}}{2} =  [ x +\dfrac { { x }^{ 3 } }{ 3! } + \dfrac {{x}^{5}}{5!} + .....] $
So, 
$ \dfrac {{e}^{\mu} - { e }^{-\mu}}{2} =  [ x +\dfrac { { \mu }^{ 3 } }{ 3! } + \dfrac {{\mu}^{5}}{5!} + .....] $
put this value in equation (1),
Sum of the terms in even places = $  { e }^{-\mu} [ \mu + \dfrac {{\mu}^{3}}{3!} + \dfrac {{\mu}^{5}}{5!} + .......]$
                    =  $  { e }^{-\mu} (\dfrac {{e}^{\mu} - { e }^{-\mu}}{2}) $
                    = $ { e }^{-\mu} \sinh { \mu  } $
                    = $ { e }^{-m} \sinh {m} $       [ Variance (m) is equal to mean ($\mu$) in Poisson distribution ]

If m is the variance of P.D., then the ratio of sum of the terms in odd places to the sum of the terms in even places is

  1. $e^{-m}\cosh m$

  2. $e^{-m}\sinh m$

  3. $\coth m$

  4. $\tanh m$


Correct Option: C
Explanation:

If m is the variance of P. D, then  P$(x; \mu) = \frac { { e }^{-\mu}{\mu}^{x}}{x!} $
Sum of the terms in odd places = [ P$(0; \mu) + P(2; \mu)  + P(4; \mu) + .........$ ]


                                       =  [$\dfrac { { e }^{-\mu}{\mu}^{0}}{0!} + \dfrac { { e }^{-\mu}{\mu}^{2}}{2!} + \dfrac { { e }^{-\mu}{\mu}^{4}}{4!} + ....... $]

                                      =  $  { e }^{-\mu} [ 1 + \dfrac {{\mu}^{2}}{2!} + \dfrac {{\mu}^{4}}{4!} + .......]$       --------------------- (1)

  Since   ${ e }^{ x }=1+\dfrac { x }{ 1! } +\dfrac { { x }^{ 2 } }{ 2! } +\dfrac { { x }^{ 3 } }{ 3! } +\dfrac { { x }^{4} }{ 4! } +....$

    ${ e }^{ -x }=1+\dfrac {( -x )}{ 1! } +\dfrac { { (-x) }^{ 2 } }{ 2! } +\dfrac { {( -x) }^{ 3 } }{ 3! } +\dfrac { { (-x) }^{4} }{ 4! } +....$ 

on adding , we get  
 ${e}^{x} + { e }^{-x} =  2[ 1 + \dfrac {{x}^{2}}{2!} + \dfrac {{x}^{4}}{4!} + .....] $
$ \dfrac {{e}^{x} + { e }^{-x}}{2} =  [ 1 +\dfrac { { x }^{ 2} }{ 2! } + \dfrac {{x}^{4}}{4!} + .....] $
So, 
$ \dfrac {{e}^{\mu} + { e }^{-\mu}}{2} =  [ 1+\dfrac { { \mu }^{ 2 } }{ 2! } + \dfrac {{\mu}^{4}}{4!} + .....] $

put this value in equation (1),
Sum of the terms in odd places = $  { e }^{-\mu} [ 1 + \dfrac {{\mu}^{2}}{2!} + \dfrac {{\mu}^{4}}{4!} + .......]$
                    =  $  { e }^{-\mu} (\dfrac {{e}^{\mu} + { e }^{-\mu}}{2}) $
                    = $ { e }^{-\mu} \cosh { \mu  } $
                    = $ { e }^{-m} \cosh {m} $       [ Variance (m) is equal to mean ($\mu$) in Poisson distribution ]
Similarly Sum of the terms in even places =  $ { e }^{-m} \sinh {m} $ 
The ratio of sum of the terms in odd places to the sum of the terms in even places is = $ \dfrac { { e }^{ -m }\cosh { m }  }{ { e }^{ -m }\sinh { m }  } =\coth { m }   $ 

A : the sum of the times in odd places in a P.D is $e^{-\lambda }$ cosh $\lambda$ 
R : cosh $\lambda =\frac{\lambda ^{1}}{1!}+\frac{\lambda ^{3}}{3!}+\frac{\lambda ^{5}}{5!}+......$

  1. Both A and R are true and R is the correct

    explanation of A

  2. Both A and R are true but R is not correct

    explanation of A

  3. A is true but R is false

  4. A is false but R is true


Correct Option: C
Explanation:

Considering all odd places in Poisson's Distribution, we get 
$e^{-\lambda}+\dfrac{e^{-\lambda}.\lambda^{2}}{2!}+\dfrac{e^{-\lambda}.\lambda^{4}}{4!}....$

$=e^{-\lambda}[1+\dfrac{\lambda^{2}}{2!}+\dfrac{\lambda^{4}}{4!}+....]$

$=e^{-\lambda}[\dfrac{e^{-\lambda}+e^{\lambda}}{2}]$

$=e^{-\lambda}(\cosh\lambda)$.
Hence reason is false.

If $X$ is a poisson variate with $P(X=0)=P(X=1)$, then $P(X=2)$ is

  1. $\dfrac{e}{2}$

  2. $\dfrac{e}{6}$

  3. $\dfrac{1}{6e}$

  4. $\dfrac{1}{2e}$


Correct Option: D
Explanation:

Here,  $P(X=0)=P(X=1)$
$\Rightarrow \displaystyle \frac{e^{-\lambda}\lambda^0}{0!}=\frac{e^{-\lambda}\lambda^1}{1!}\Rightarrow \lambda = 1$
$\displaystyle \therefore P(X = 2) = \frac{e^{-\lambda}1^2}{2!}=\frac{1}{2e}$

If $X$ is a random poisson variate such that $E(X^{2})=6$, then $E(X)=$

  1. $-3$

  2. $2$

  3. $-3&amp;2$

  4. $-2$


Correct Option: B
Explanation:

Variance  = $E(X^{ 2 }) - E[X]^{ 2 }$
Mean = Variance for P.D.
Let mean = $m$
Therefore
$m = $ $ 6 - m*m $
Hence
$m = 2$

For a Poisson variate $X$ if $P(X=2)=3P(X=3)$, then the mean of $X$ is

  1. $1$

  2. $1/2$

  3. $1/3$

  4. $1/4$


Correct Option: A
Explanation:

Poisson's distribution implies
$P(X=n)=\dfrac{\lambda^{n}e^{-\lambda}}{n!}$
Where mean=variance=$\lambda$
Hence $P(x=2)=3P(x=3)$
$3\dfrac{\lambda^{3}e^{-\lambda}}{3!}=\dfrac{\lambda^{2}e^{-\lambda}}{2!}$
$3\lambda=3$
$\lambda=1$
Hence mean=variance=$1$

If $X$ is a poisson variate such that $P(X=0)=\dfrac{1}{2}$, the variance of $X$ is

  1. $\dfrac{1}{2}$

  2. $2$

  3. $\log _{e}2$

  4. $3$


Correct Option: C
Explanation:

Poisson's distribution implies
$P(X=n)=\frac{\lambda^{n}e^{-\lambda}}{n!}$
Where mean=variance=$\lambda$
Hence substituting $x=0$ 

we get
$e^{-\lambda}=0.5$
$e^{\lambda}=2$
$\lambda=ln(2)$
Hence mean=variance=$log _{e}(5)$

If in a poisson frequency distribution, the frequency of $3$ successes is $\displaystyle \frac{2}{3}$ times the frequency of $4$ successes, the mean of the distribution is

  1. $\displaystyle \frac{2}{3}$

  2. $\displaystyle \frac{1}{3}$

  3. $6$

  4. $\sqrt{6}$


Correct Option: C
Explanation:

According to question
$P(X=3; \mu )= \dfrac {2}{3}P(X=4; \mu) $
$\dfrac { { e }^{-\mu}{\mu}^{3}}{3!}  = \dfrac {2}{3} \dfrac { { e }^{-\mu}{\mu}^{4}}{4!} $
=> $\dfrac { { e }^{-\mu}{\mu}^{3}}{6}  = \dfrac { { e }^{-\mu}{\mu}^{4}}{36} $
=> ${ e }^{ -\mu  }{ \mu  }^{ 3 }\left[ \dfrac { 1 }{ 6 } -\dfrac { \mu  }{ 36 }  \right] $ 
=> $ \mu = 0 \  or\   \mu = \dfrac{36}{6} = 6 $

If X is a poisson variate such that $P(X=2)=9p(X=4)+90p(X=6)$ , then the mean of x is

  1. $3$

  2. $2$

  3. $1$

  4. $0$


Correct Option: C
Explanation:

For Poisson's distribution, $P(X)$ is given by 
$\displaystyle P(X)=\frac { { e }^{ -\lambda  }.{ \lambda  }^{ x } }{ x! } $
It is given that $P(X=2)=9P(X=4)+  90P(X=6)$
$\Rightarrow \displaystyle \frac { { e }^{ -\lambda  }.{ \lambda  }^{ 2 } }{ 2! } =9.\frac { { e }^{ -\lambda  }.{ \lambda  }^{ 4 } }{ 4! } +90.\frac { { e }^{ -\lambda  }.{ \lambda  }^{ 6 } }{ 6! } $
Cancelling ${ e }^{ -\lambda  }$ from both sides, we get
$\displaystyle \frac { { \lambda  }^{ 2 } }{ 2! } =9.\frac { { \lambda  }^{ 4 } }{ 4! } +90.\frac { { \lambda  }^{ 6 } }{ 6! } $
Since $\lambda$ can not be zero, cancel out $\lambda^2$ from both sides to obtain
${ \lambda  }^{ 4 }+3{ \lambda  }^{ 2 }-4=0$.
Solving , we get $\lambda = 1$ as a valid solution.

If $X$ is a poisson variate such that $P(X=0)=0.1,P(X=2)=0.2$, then the parameter $\lambda $

  1. $2$

  2. $4$

  3. $5$

  4. $3$


Correct Option: A
Explanation:

$P(X=0)=0.1$ and $P(X=2)=0.2$
$P$$(0; \mu) = \dfrac{ { e }^{-\mu}{\mu}^{0}}{0!} =  { e }^{-\mu}$  =  $0.1 $
$P$$(2; \mu) = \dfrac{ { e }^{-\mu}{\mu}^{2}}{2!} $  $=  0.2 $
                    =>  $\dfrac {{0.1}{\mu}^{2}}{2} $ $= 0.2$
                    =>  $ \mu $ $= 2 $
The parameter  is mean ($ \mu $) in Poisson distribution, so $ =$ $2  $

If $X$ is a poisson variate with $P(X=0) = 0.8,$ then the variance of $X$ is

  1. $log _{e}20$

  2. $log _{10}20$

  3. $log _{e}(5/4)$

  4. $0$


Correct Option: C
Explanation:

In Poisson distribution,   $P(X=0) = 0.8,$
 $ p(x; \mu) = \dfrac { { e }^{-\mu}{\mu}^{x}}{x!} $ 
=>  $ p(0; \mu) = \dfrac { { e }^{-\mu}{\mu}^{0}}{0!} $
=>    0.8 = $ { e }^{-\mu} $ 
=>    $ \dfrac {4}{5} = { e }^{-\mu} $
=>    ${ e }^{\mu}  = \dfrac {5}{4} $ 
taking log to both sides 
 =>  $  \log _{ e }{ { e }^{ \mu  } }  = \log _{e} (\dfrac {5}{4}) $ 
 =>  $ \mu =  \log _{e} (\dfrac {5}{4}) $

If in a poisson distribution $P(X=1)=P(X=2)$; the mean of the distribution $f(x)=e^{-x}\dfrac{\lambda ^{x}}{\angle x}$ is

  1. $1$

  2. $2$

  3. $\dfrac{1}{2}$

  4. $\dfrac{3}{2}$


Correct Option: B
Explanation:

Poisson's distribution implies
$P(X=n)=\frac{\lambda^{n}e^{-\lambda}}{n!}$
Where mean=variance=$\lambda$
Hence $P(x=1)=P(x=2)$
$\frac{\lambda^{1}e^{-\lambda}}{1!}=\frac{\lambda^{2}e^{-\lambda}}{2!}$
$\lambda=2$
Hence mean=variance=$2$

If for a poisson distribution $P(X=0)=0.2$, then the variance of the distribution is

  1. $5$

  2. $log _{10}5$

  3. $log _{e}5$

  4. $log _{5}e$


Correct Option: C
Explanation:

Poisson's distribution implies
$P(X=n)=\frac{\lambda^{n}e^{-\lambda}}{n!}$
Where mean=variance=$\lambda$
Hence substituting $x=0$ we get
$e^{-\lambda}=0.2$
$e^{\lambda}=5$
$\lambda=ln(5)$
Hence mean=variance=$log _{e}(5)$

In a Poisson distribution, the probability $P(X=0)$ is twice the probability $P(X=1)$. The mean of the distribution is

  1. $\displaystyle \frac{1}{4}$

  2. $\displaystyle \frac{1}{3}$

  3. $\displaystyle \frac{1}{2}$

  4. $\displaystyle \frac{3}{4}$


Correct Option: C
Explanation:

Poisson's distribution implies
$P(X=n)=\dfrac{\lambda^{n}e^{-\lambda}}{n!}$
Where mean=variance=$\lambda$
Hence $P(x=0)=2P(x=1)$
$\dfrac{\lambda^{0}e^{-\lambda}}{0!}=2\dfrac{\lambda^{1}e^{-\lambda}}{1!}$
$2\lambda=1$
$\lambda=\dfrac{1}{2}$
Hence mean=variance=$0.5$

Suppose $X$ is a poisson variable such that $P(X=2)=\frac{2}{3}P(X=1)$, then $P(x=0)$ is

  1. $\dfrac{3}{4}$

  2. $e^{\dfrac{4}{3}}$

  3. $e^{\dfrac{-4}{3}}$

  4. $\dfrac{1}{2}$


Correct Option: C
Explanation:

Poisson's distribution implies
$P(X=n)=\dfrac{\lambda^{n}e^{-\lambda}}{n!}$

Where mean=variance=$\lambda $
Hence, $P(X=2)=\dfrac{2}{3}P(x=1)$
$\dfrac{\lambda^{2}e^{-\lambda}}{2!}=\dfrac{2\lambda^{1}e^{-\lambda}}{3.1!}$
$\dfrac{\lambda}{2}=\dfrac{2}{3}$
$\lambda=\dfrac{4}{3}$
Hence, $P(X=0)=e^{\dfrac{-4}{3}}$

If $X$ is a poisson variable such that $P(X=2)=\frac{2}{3}P(X=1)$, then $P(x=3)$ is

  1. $e^{\frac{-4}{3}}$

  2. $\frac{64}{162}e^{\frac{-4}{3}}$

  3. $e^{\frac{-3}{4}}$

  4. $e^{\frac{3}{4}}$


Correct Option: B
Explanation:

$P(X=2)= \cfrac {2}{3}P(X=1) $

$P$$(2; \mu) = \cfrac { { e }^{-\mu}{\mu}^{2}}{2!} = \cfrac {2}{3} P(1; \mu) $=$ \cfrac {2}{3} \cfrac { { e }^{-\mu}{\mu}^{1}}{1!}$

$ { e }^{-\mu} \mu (\cfrac {\mu}{2} - \cfrac {2}{3})=0$

either $\mu = 0\  or \mu = \cfrac{4}{3}$

but here $ \mu = \cfrac{4}{3}$$P(X=3) =$ $ \cfrac { { e }^{-\mu}{\mu}^{3}}{3!} = \cfrac { { e }^{ \frac { -4 }{ 3 }  }{( \frac { 4 }{ 3 }  })^{ 3 } }{ 6 } = \cfrac { 64 }{ 162 } { e }^{ \frac{-4}{3} } $

If $X$ is a Poisson variate with parameter $1.5$, then $P(X>1)$ is

  1. $1-e^{-1.5}$

  2. $e^{-1.5}(2.5)$

  3. $1-e^{-1.5}(2.5)$

  4. $1-e^{-1.5}(3.5)$


Correct Option: C
Explanation:

$P(X>1)$
$=1-[P(X=0)+P(X=1)]$
$=1-[e^{-1.5}+1.5e^{-1.5}]$
$=1-e^{1.5}(1+1.5)$
$=1-e^{1.5}(2.5)$

If $X$ is a poisson variate such that $P(X=0)=P(X=1)$,then $P(X=2)=$

  1. $\dfrac{e}{2}$

  2. $\dfrac{e}{6}$

  3. $\dfrac{1}{6e}$

  4. $\dfrac{1}{2e}$


Correct Option: D
Explanation:

$P(X=0)=P(X=1) $
P$(0; \mu) = \dfrac { { e }^{-\mu}{\mu}^{0}}{0!} =  P(1; \mu) = \dfrac { { e }^{-\mu}{\mu}^{1}}{1!}$
$ { e }^{-\mu}  (1 - \mu )$ = 0 
 $\mu = 1 $
P(X=2) = $ \dfrac { { e }^{-\mu}{\mu}^{2}}{2!} = \dfrac{{ e }^{-1}}{2} = \dfrac {1}{2e} $

A random variable $X$ follows poisson distribution such that $P(X=k)=P(X=k+1)$ then the parameter of the distribution $\lambda =$

  1. $K$

  2. $K+1$

  3. $\dfrac{K}{2}$

  4. $\dfrac{K+1}{2}$


Correct Option: B
Explanation:

$P(X=k)=P(X=k+1)$
$ P(x;\mu)=\dfrac { { e }^{ -\mu }{ \mu }^{ x } }{ x! } $


$ P(k;\mu)=\dfrac { { e }^{ -\mu }{ \mu }^{ k } }{ k! } =P(k+1;\mu)=\dfrac { { e }^{ -\mu }{ \mu }^{ k+1 } }{ (k+1)! }$

$ 1 = \dfrac {\mu}{k+1} $

$ \mu = k+1 $

In a poisson distribution $P(X=0)=P(X=1)=k$, then the value of $k$ is

  1. $1$

  2. $\displaystyle\frac{1}{e}$

  3. $e$

  4. $\sqrt{2}$


Correct Option: B
Explanation:

$P(x;\mu)=\dfrac { { e }^{ -\mu }{ \mu }^{ x } }{ x! }$

Given,$P(X=0)=P(X=1)=k $

$ \dfrac { { e }^{ -\mu }{ \mu }^{ 0 } }{ 0! } =\dfrac { { e }^{ -\mu }{ \mu }^{ 1 } }{ 1! }  $
$ { e }^{ -\mu } ={ e }^{ -\mu } \mu $
$ { e }^{ -\mu } (1 - \mu) = 0 $ 
Since $ { e }^{ -\mu } \neq 0$ , => $ \mu = 1 $
$ P(X=0) = \dfrac { { e }^{ -\mu }{ \mu }^{ 0 } }{ 0! } = { e }^{ -\mu } ={ e }^{ - 1} = \dfrac {1}{e} $

If for a poisson variable $ X$, $P(X=1)=2.\ P(X=2)$, then the parameter $\lambda $ is

  1. $0$

  2. $1$

  3. $2$

  4. $3$


Correct Option: B
Explanation:

For Poisson's distribution, $P(X)$ is given by 
$\displaystyle P(X)=\frac { { e }^{ -\lambda  }.{ \lambda  }^{ x } }{ x! } $
Hence $\displaystyle P(X=1) =\displaystyle  \frac { { e }^{ -\lambda }.{ \lambda  }^{ 1 } }{ 1! } $.
Similarly $\displaystyle P(X=2)=\frac { { e }^{ -\lambda }.{ \lambda  }^{ 2 } }{ 2! } $.
Given that, $P(X=1) = 2 P(X=2)$
$\Rightarrow \displaystyle \frac { { e }^{ -\lambda }.{ \lambda  }^{ 1 } }{ 1! } =2.\frac { { e }^{ -\lambda }.{ \lambda  }^{ 2 } }{ 2! } $
Simplifying we get $\lambda = 1$

If $X$ is a Poisson variate such that $P(X=1) = P(X=2)$ then $P(X=4)=$

  1. $\dfrac{1}{2e^{2}}$

  2. $\dfrac{1}{3e^{2}}$

  3. $\dfrac{2}{3e^{2}}$

  4. $\dfrac{1}{e^{2}}$


Correct Option: C
Explanation:

$P(x;\mu)=\dfrac { { e }^{ -\mu }{ \mu }^{ x } }{ x! }$
$P(X=1) = P(X=2)$


$\dfrac { { e }^{ -\mu }{ \mu }^{ 1 } }{ 1! } =\dfrac { { e }^{ -\mu }{ \mu }^{ 2 } }{ 2! } $

$ \mu = 2 $
$ P(X=4)= \dfrac { { e }^{ -2 }{ 2 }^{ 4 } }{ 4! }= \dfrac {2}{3{e}^{2}} $

If a random variable $X$ follows a P.D. such that $P(X=1)=P(X=2)$, then $P(X=0)=$

  1. $e^{2}$

  2. $\dfrac{1}{e^{2}}$

  3. $\dfrac{1}{e}$

  4. $e$


Correct Option: B
Explanation:

$ P(X=1)=P(X=2)$
 $P$$(1; \mu) = \dfrac { { e }^{-\mu}{\mu}^{1}}{1!} =  P(2; \mu) = \dfrac { { e }^{-\mu}{\mu}^{2}}{2!}$
$ { e }^{-\mu} [\mu (2 - \mu )]$ $= 0$

either $\mu = 0  or  \mu = 2 $
$P(X=0) = $$ \dfrac { { e }^{-\mu}{\mu}^{0}}{0!} = { e }^{-\mu}$

So, either $ { e }^{-0}$or ${ e }^{-2}$
that is 1 or $\dfrac {1}{{ e }^{2}} $

If the first two terms of a Poisson distribution are equal to $k$, find $k$.

  1. $e$

  2. $\displaystyle \frac{1}{e}$

  3. $1$

  4. $2$


Correct Option: B
Explanation:

Poisson's distribution implies
$P(X=n)=\dfrac{\lambda^{n}e^{-\lambda}}{n!}$
Where mean=variance=$\lambda$
Hence $P(x=0)=P(x=1)$
$\dfrac{\lambda^{0}e^{-\lambda}}{0!}=\dfrac{\lambda^{1}e^{-\lambda}}{1!}$
$\lambda=1$
Hence $P(X=0)=P(X=1)$
$=\dfrac{1}{e}$

In a binomial distribution $n = 200, p = 0.04$. Taking Poisson distribution as an approximation to the binomial distribution .
Assertion (A) :- Mean of the Poisson distribution $= 8$
Reason (R) : In a Poisson distribution, $\displaystyle P(X=4)=\frac{512}{3e^{8}}$

  1. both A and R are true and R is the correct explanation of A

  2. both A and R are true and R is not correct explanation of A

  3. A is true but R is false

  4. A is false but R is true


Correct Option: B
Explanation:

mean of P.D. $=np = 8$
And $P(X = 4) = \cfrac{e^{-8}(8)^4}{4!}=\cfrac{512}{3e^8}$
Hence both statement are correct but they are not related to each other.

If $X$ is a random poission variate such that $2P(X=0)+P(X=2)=2P(X=1)$ then $E(X)=$

  1. $4$

  2. $3$

  3. $2$

  4. $1$


Correct Option: C
Explanation:

Poisson's distribution implies
$P(X=n)=\dfrac{\lambda^{n}e^{-\lambda}}{n!}$
Where mean=variance=$\lambda$
Hence $2.P(x=0)+P(x=2)=2P(x=1)$
$2\dfrac{\lambda^{0}e^{-\lambda}}{0!}+\dfrac{\lambda^{2}e^{-\lambda}}{2!}=2\dfrac{\lambda^{1}e^{-\lambda}}{1!}$


$2e^{-\lambda}+\dfrac{\lambda^{2}e^{-\lambda}}{2!}=2\lambda: e^{-\lambda}$

$2\lambda: e^{-\lambda}-2e^{-\lambda}=\dfrac{\lambda^{2}e^{-\lambda}}{2!}$
$4\lambda-4=\lambda^{2}$
$\lambda^{2}-4\lambda+4=0$
$(\lambda-2)^{2}=0$
$\lambda=2$
Hence mean=variance=$2$

If the probability of that a poisson variable $X$ takes a positive value $\geq 1$ is $1-e^{-1.5}$, then the varianceof the distribution is

  1. $4$

  2. $3$

  3. $1.5$

  4. $0$


Correct Option: C
Explanation:

Given $P(X\geq 1) = 1-e^{-1.5}$
$\Rightarrow 1-P(X=0)=1-e^{-1.5}\Rightarrow P(X=0)=e^{-1.5}=e^{-\lambda}\therefore \lambda = 1.5$

In a town $10$ accidents take place in a span of $50$ days. Assuming that number of accidents follows Poisson distribution, the probability that there will be atleast one accident on a selected day at random is

  1. $\displaystyle \frac{e^{-0.02}.2^{1}}{1!}$

  2. $1-e^{-0.2}$

  3. $e^{-0.2}$

  4. $1-e^{1.2}$


Correct Option: B
Explanation:

Here Poisson parameter $\lambda = \cfrac{10}{50}=0.2$
$\therefore P(x \geq 1)=1-P(X=0)=1-\cfrac{e^{-0.2}.(.2)^0}{0!}=1-e^{-0.2}$ 

A car hire firm has $2$ cars which it hires out day by day. If the number of demands for a car on each day follows Poisson distribution with parameter $1.5$, then the probability that both the cars is used is

  1. $1.12 \times e^{-1.5}$

  2. $1-2.5 \times e^{-1.5}$

  3. $1-3.625 \times e^{-1.5}$

  4. $3.625 \times e^{-1.5}$


Correct Option: B
Explanation:

Here $\lambda = 1$
Hence probability that both the cars are used $=1-P(X=0)-P(X=1)=1-\cfrac{e^{-1.5}(1.5)^0}{0!}-\cfrac{e^{-1.5}(1.5)^1}{1!}=1-2.5e^{-1.5}$

If $X$ is a Poisson variate with parameter $\displaystyle \frac{3}{2}$, find $P(X\geq 2)$

  1. $\displaystyle \frac{5}{2}e^{\frac{-3}{2}}$

  2. $\displaystyle 1-\frac{5}{2}e^{\frac{-3}{2}}$

  3. $\displaystyle 1-e^{\frac{-3}{2}}$

  4. $\displaystyle e^{\frac{-3}{2}}$


Correct Option: B
Explanation:

Here $\lambda = \cfrac{3}{2}$
$\therefore P(X\geq 2) = 1-P(X=0)-P(X=1)$
$\displaystyle =1-\cfrac{e^{-3/2}(3/2)^0}{0!}-\cfrac{e^{-3/2}(3/2)^1}{1!}=1-\cfrac{5}{2}e^{-3/2}$

If $X$ is a random Poisson variate such that $P(X=0)=\displaystyle\frac{1}{e}$, then the variance of the same distribution is

  1. $1$

  2. $2$

  3. $3$

  4. $4$


Correct Option: A
Explanation:

Poisson's distribution implies
$P(X=n)=\dfrac{\lambda^{n}e^{-\lambda}}{n!}$
Where mean=variance=$\lambda$
Hence $P(x=0)=\frac{1}{e}$
$\dfrac{\lambda^{0}e^{-\lambda}}{0!}=\dfrac{1}{e}$
$\lambda=1$
Hence mean=variance=$1$

If on an average ,5 percent of the output in a factory making certain parts, is defective and that 200 units are in a package then the probability that atmost 4 defective parts may be found in that package is

  1. $\displaystyle e^{-10}\left [ 1+\frac{100}{1!}+\frac{100^{2}}{2!}+\frac{100^{3}}{3!}+\frac{100^{4}}{4!} \right ]$

  2. $\displaystyle e^{-10}\left [ 1+\frac{10}{1!}+\frac{10^{2}}{2!}+\frac{10^{3}}{3!}+\frac{10^{4}}{4!} \right ]$

  3. $\displaystyle e^{-10}\left [ 1-\frac{10}{1!}+\frac{10^{2}}{2!}+\frac{10^{3}}{3!}+\frac{10^{4}}{4!} \right ]$

  4. $\displaystyle e^{-10}\left [ 1-\frac{100}{1!}+\frac{100^{2}}{2!}+\frac{100^{3}}{3!}+\frac{100^{4}}{4!} \right ]$


Correct Option: B
Explanation:

Here $\lambda$
$=\dfrac{5}{100}.200$
$=10$.
Hence by applying Poisson distribution, we get that the probability that atmost 4 defective part are found is 
$=\sum _{k=0} ^{k=4} \dfrac{e^{-10}.10^{k}}{k!}$

$=e^{-10}[1+\dfrac{10}{1!}+\dfrac{10^{2}}{2!}+\dfrac{10^{3}}{3!}+\dfrac{10^{4}}{4!}]$.

Suppose $300$ misprints are distributed randomly throughout a book of $500$ pages. The probability that a given page contains, at least one misprint is 

  1. $1.e^{-0.6}$

  2. $1-e^{-0.6}$

  3. $(0.6)e^{-0.6}$

  4. $(0.06)e^{-0.6}$


Correct Option: B
Explanation:

Here $\lambda = \cfrac{300}{500}=0.6$
Hence $ P(X\geq 1) = 1-P(X=0)=1-\cfrac{e^{-0.6}(0.6)^0}{0!}=1-e^{-0.6}$

A manufactured product on an average has 2 defects per unit of product produced. If the number of defects follows Poisson distribution, the probability of finding at least one defect is 

  1. $e^{-2}$

  2. $1-e^{-2}$

  3. $\displaystyle \frac{e^{-2}2^{1}}{1!}$

  4. $e^{-0.02}$


Correct Option: B
Explanation:

By using Poisson distribution, 
$ P(x;\mu)=\dfrac { { e }^{ -\mu }{ \mu }^{ x } }{ x! } $
$ \mu $: The mean number of successes that occur in a specified region.
$x$: The actual number of successes that occur in a specified region.
$P(x$; $ \mu $): The Poisson probability that exactly x successes occur in a Poisson experiment, when the mean number of successes is $ \mu $
$ P(x \ge 1; \mu) = 1 - P (x=0 ; \mu) $
$ \mu = 2 $    
$x = 0$ (No defective product) 
$ P(x \ge 1; 2) = 1 - P (x=0 ; 2) = 1 - \dfrac { { e }^{ -2 }{ 2 }^{ 0} }{ 0! } = 1 - {e}^{-2}$
                             

A car hire firm has $2$ cars which it hires out day by day. If the number of demands for a car on each day follows poisson distribution with parameter $1.5$, then the probability that only one car is used is

  1. $e^{-1.5}$

  2. $1.5\times e^{-1.5}$

  3. $1-2.5\times e^{-1.5}$

  4. $1-1.5\times e^{-1.5}$


Correct Option: B
Explanation:

Here $\lambda = 1.5$
Hence probability that only one car is used is $=P(X=1) = \cfrac{e^{-1.5}(1.5)}{1!}=1.5e^{-1.5}$

If $3$% of electric bulbs manufactured by a company are defective, the probability that a sample of $100$ bulbs has no defective bulbs is

  1. 0

  2. $e^{-3}$

  3. $1-e^{-3}$

  4. $3e^{-3}$


Correct Option: B
Explanation:

By using Poisson distribution, 
$ P(x;\mu)=\dfrac { { e }^{ -\mu }{ \mu }^{ x } }{ x! } $
$ \mu $: The mean number of successes that occur in a specified region.
x: The actual number of successes that occur in a specified region.
P(x;$ \mu $ ): The Poisson probability that exactly x successes occur in a Poisson experiment, when the mean number of successes is $ \mu $
$ \mu = 100 \times 0.03 = 3 $    
$x = 0$ (No defective bulbs) 
$ P(0; 3)=\dfrac { { e }^{ -3 }{ 3 }^{ 0 } }{ 0! }  $
$P(0;3)={ e }^{ -3 } $

On an average, a submarine on patrol sights $6$ enemy ships per hour. Assuming the number of ships sighted in a given length of time is a poisson variate, the probability of sighting atleast one ship in the next $15$ minutes is

  1. $e^{-15}$

  2. $1-e^{-6}$

  3. $1-e^{-15}$

  4. $e^{-6}$


Correct Option: C
Explanation:

The probability of seeing atleast one ship
=1-(probability of seeing no ship)
$=1-\dfrac{\lambda^{0}.e^{-\lambda}}{0!}$
$=1-e^{-\lambda}$
It is given the Poisson's variate is the number of ships passing per unit time.
Hence in the above case $\lambda=15$
Thus the required probability is
$=1-e^{-15}$.

If the number of telephone calls coming into a telephone exchange between 10 AM and 11 AM follows P.D. with parameter 2, then the probability of obtaining zero calls in that time interval is

  1. $e^{-2}$

  2. $1-e^{-2}$

  3. $2.e^{-2}$

  4. $3.e^{-2}$


Correct Option: A
Explanation:

Here P.D parameter $\lambda = 2$
Hence probability of obtaining zero calls during 10 AM to 11 AM is $=(P(X=0)=e^{-2}$ 

A manufactured product on an average has $2$ defects per unit of product produced. If the number of defects follows P.D., the probability of finding zero defects is

  1. $e^{-2}$

  2. $1-e^{-2}$

  3. $\displaystyle \frac{e^{-2}2^{1}}{\angle 1}$

  4. $e^{-002}$


Correct Option: A
Explanation:

By using Poisson distribution, 
$ P(x;\mu)=\dfrac { { e }^{ -\mu }{ \mu }^{ x } }{ x! } $
: The mean number of successes that occur in a specified region.
x: The actual number of successes that occur in a specified region.
P(x; ): The Poisson probability that exactly x successes occur in a Poisson experiment, when the mean number of successes is 
$ \mu = 2$ (defects per unit of product produced) 
x = 0 (zero defects)
$ P(0; 2)=\dfrac { { e }^{ -2 }{ 2 }^{ 0 } }{ 0! } = {e}^{-2}$

If the number of telephone calls coming into a telephone exchange between 10 AM and 11 AM follows Poisson distribution with parameter 2 then the probability of obtaining at least one call in that time interval is 

  1. $e^{-2}$

  2. $(1-e^{-2})$

  3. $2e^{-2}$

  4. $3e^{-2}$


Correct Option: B
Explanation:

By using Poisson distribution, 
$ P(x;\mu)=\dfrac { { e }^{ -\mu }{ \mu }^{ x } }{ x! } $
$ \mu $: The mean number of successes that occur in a specified region(Parameter)
x: The actual number of successes that occur in a specified region.
P(x;$ \mu $ ): The Poisson probability that exactly x successes occur in a Poisson experiment, when the mean number of successes is $ \mu $
$ P(x \ge 1; \mu) = 1 - P (x=0 ; \mu) $
$ \mu = 2 $    
$x = 0$ (No call comes) 
$ P(x \ge 1; 2) = 1 - P (x=0 ; 2) = 1 - \dfrac { { e }^{ -2 }{ 2 }^{ 0} }{ 0! } = 1 - {e}^{-2}$
                             

Cycle tyres are supplied in lots of $10$ and there is a chance of $1$ in $500$ to be defective. Using poisson distribution, the approximate number of lots containing no defectives in a consignment of $10,000$ lots if $e^{-0.02}=0.9802$ is

  1. $9980$

  2. $9998$

  3. $9802$

  4. $9982$


Correct Option: C
Explanation:

Here $\lambda = \cfrac{1}{500}\times 10 = 0.02$
Thus probability that  lot is not defective is $=P(X=0)=\cfrac{e^{-0.002}(.0020^0}{0!}=e^{-.002}=0.9802$
Hence number of no defective lots out of $10,000$ is $=.9802\times 10,000=9802$

The chance of a traffic accident in a day attributed to a taxi driver is $0.001$. Out of a total of $1000$ days the number of days with no accident is

  1. $1000\times e^{-1}$

  2. $1000\times e^{-0.1}$

  3. $1000\times e^{-0.001}$

  4. $1000\times e^{-0.0001}$


Correct Option: C
Explanation:

Here $\lambda = 0.001$
Hence number of day out of 1000 days without accident is $1000\times P(X=0)=1000\times e^{-0.001}$

A manufacturer of cotter pins knows that $5$% of his product is defective. If he sells cotter pins in boxes of $100$ and guarantees that not more than $10$ pins will be defective, the approximate probability that a box will fail to meet the guaranteed quality is

  1. $\displaystyle \frac{e^{-5}5^{10}}{ 10!}$

  2. $1-\displaystyle \sum _{x=0}^{10}\frac{e^{-5}5^{x}}{ x!}$

  3. $1-\displaystyle \sum _{x=0}^{\infty }\frac{e^{-5}5^{x}}{ x!}$

  4. $\displaystyle \sum _{x=0}^{\infty }\frac{e^{-5}5^{x}}{ x!}$


Correct Option: B
Explanation:

we are given $n=100$
let $p=$ probability of a defective bulb $=5$%$=0.05$
$\therefore m=$ mean number of defective bulbs in a box of $100=np=100\times 0.05=5$
Since p is small , we can use poison's distribution.
Probability of $x$ defective bulbs in a box of $100$ is
$\displaystyle P\left( X=x \right) =\frac { { e }^{ -m }{ m }^{ x } }{ x! } =\frac { { e }^{ -5 }{ 5 }^{ x } }{ x! } ,x=0,1,2...$
Probability that is box will fail to meet the guarented quality is $\displaystyle P\left( X>10 \right) =1-P\left( X\le 10 \right) =1-\sum _{ x=0 }^{ 10 }{ \frac { { e }^{ -5 }{ 5 }^{ x } }{ x! }  } =1-{ e }^{ -5 }\sum _{ x=0 }^{ 10 }{ \frac { { 5 }^{ x } }{ x! }  } $

The number of accidents in a year attributed to a taxi driver in a city follows Poisson distribution with mean $3$. Out of $1000$ taxi drivers, the approximate number of drivers with no accident in a year given that $e^{-3}=0.0498$ is

  1. $4.98$

  2. $49.8$

  3. $498$

  4. $4.8$


Correct Option: B
Explanation:

Here $\lambda = 3$
Hence Number of drivers with no accident out of 1000 is $=1000\times P(X=0)=1000\times e^{-3}=1000\times 0.0498=49.8$

A manufacturing concern employing a large number of workers finds that, over a period of time, the average absentee rate is $2$ workers per shift. The probability that exactly $2$ workers will be absent in a chosen shift at random is

  1. $\displaystyle \frac{e^{-2}2^{2}}{ 2!}$

  2. $\displaystyle \frac{e^{-2}2^{3}}{3!}$

  3. $e^{-2}$

  4. $e^{-3}$


Correct Option: A
Explanation:

By using Poisson distribution, 
$ P(x;\mu)=\dfrac { { e }^{ -\mu }{ \mu }^{ x } }{ x! } $
$ \mu $: The mean number of successes that occur in a specified region(parameter)
x: The actual number of successes that occur in a specified region.
P(x; $ \mu $): The Poisson probability that exactly x successes occur in a Poisson experiment, when the mean number of successes is $ \mu $
Here,$ \mu $  = 2 
x = 2 (exact 2 workers)
$ P(2;2)=\dfrac { { e }^{ -2 }{ 2 }^{2 } }{ 2! } $

A manufacturer who produces medicine bottles finds that $0.1$% of the bottles are defective. The bottles are packed in boxes containing $500$ bottles. A drug manufacturer buys $100$ boxes from the producer of bottles. Using poisson distribution,the number of boxes with at least one defective bottle is

  1. $100(1-e^{-0.1})$

  2. $100(1-e^{-0.5})$

  3. $100(1-e^{-0.05})$

  4. $100(1-e^{-0.01})$


Correct Option: B
Explanation:

Here $\lambda = \cfrac{0.1}{100}\times 500=0.5 $
Hence number of boxes out of 100 which contain at least one defective bottle is,
$=100\left(1-P(X=0)\right)=100\left(1-e^{-0.5}\right)$ 

Suppose $2$% of the people on an average are left handed. The probability of 3 or more left handed among 100 people is

  1. $3e^{-2}$

  2. $4e^{-2}$

  3. $1-5e^{-2}$

  4. $5 e^{-2}$


Correct Option: C
Explanation:

Here P.D parameter $\lambda = \cfrac{2}{100}\times 100=2$
Hence probability that 3 or more people are left handed is $=1-P(X=0)-P(X=1)-P(X=2)$
$=1-e^{-2}-\cfrac{e^{-2}2}{1!}-\cfrac{e^{-2}2^2}{2!}=1-5e^{-2}$

Suppose there is an average of $2$ suicides per year per $50,000$ population. In a city of population $1,00,000$, the probability that in a given year there are, zero suicides is

  1. $1.e^{-2}$

  2. $1-e^{-2}$

  3. $e^{-4}$

  4. $1-e^{-4}$


Correct Option: C
Explanation:

P.D parameter $\lambda = \cfrac{2}{50000}\times 100000 = 4$
Thus probability that in a year there is no suicide is $=P(X=0)=e^{-4}$

Suppose on an average $5$ out of $2000$ houses get damaged due to fire accident during summer. Out of $10,000$ houses in a locality, the probability that exactly $10$ houses will get damaged during summer is

  1. $\displaystyle \frac{e^{-5}5^{10}}{ 10!}$

  2. $\displaystyle \frac{e^{-10}10^{10}}{ 10!}$

  3. $\displaystyle \frac{e^{-25}25^{10}}{10!}$

  4. $\displaystyle \frac{e^{-15}15^{10}}{10!}$


Correct Option: C
Explanation:

Here P.D parameter $\lambda = \cfrac{5}{2000}\times 10000=25$
Hence probability that exactly 10 houses will get damaged $=P(X = 10) = \cfrac{e^{-25}(25)^{10}}{10!}$

A manufacturer who produces medicine bottles finds that $0.1$$\%$ of the bottles are defective. The bottles are packed in boxes containing $500$ bottles. A drug manufacturer buys $100$ boxes from the producer of bottles. Using Poisson distribution, the number of boxes with no defective bottle is

  1. $100\times e^{-0.1}$

  2. $100\times e^{-0.5}$

  3. $100\times e^{-0.05}$

  4. $100\times e^{-0.01}$


Correct Option: B
Explanation:

$\displaystyle p=\frac { 0.1 }{ 100 } =0.001,n=500\ \lambda =np=500\times 0.001=0.5\ N=100$
We know that $\displaystyle p\left( r \right) =e^{ -1 }\frac { { \lambda  }^{ r } }{ r! } $
Number of boxes containing no defective bottle 
$\displaystyle =N.P.\left( r=0 \right) =1000\times { e }^{ 0.5 }\frac { \left( 0.5 \right) ^{ 0 } }{ 0! } =1000\times { e }^{ -0.5 }$

A company knows on the basis of past experience that $2$% of its blades are defective. The probability of having $3$ defective blades in a sample of $100$ blades if $e^{-2}=0.1353$ is

  1. $0.1353$

  2. $0.1804$

  3. $0.2706$

  4. $0.3606$


Correct Option: B
Explanation:

Here P.D parameter $\lambda = \cfrac{2}{100}\times 100=2$
Hence number of probability that 3 blade are defective is $=P(X=3)=\cfrac{e^{-2}(2)^3}{3!}=\cfrac{4}{3}e^{-2}=\cfrac{4}{3}\times .1353=.1804$

On the average a submarine on patrol sights $6$ enemy ships per hour. Assuming the number of ships sighted in a given length of time is a poisson variate, the probability of sighting $4$ ships in the next two hours is

  1. $\displaystyle \frac{e^{-12}12^{4}}{ 4!}$

  2. $\displaystyle \frac{e^{-4}12^{12}}{ 3!}$

  3. $\displaystyle \frac{e^{-6}12^{4}}{ 4!}$

  4. $\displaystyle \frac{e^{-3}12^{2}}{ 4!}$


Correct Option: A
Explanation:

By using Poisson distribution, 
$ P(x;\mu)=\dfrac { { e }^{ -\mu }{ \mu }^{ x } }{ x! } $
$ \mu $: The mean number of successes that occur in a specified region(parameter)
x: The actual number of successes that occur in a specified region.
P(x; $ \mu $): The Poisson probability that exactly x successes occur in a Poisson experiment, when the mean number of successes is $ \mu $
the average a submarine on patrol sights 6 enemy ships per hour, so for 2 hours
Here, $ \mu =  6 \times 2 = 12 $
x = 4 ( ships)
$ P(4;12)=\dfrac { { e }^{ -12 }{12 }^{4 } }{ 4! }$

Patients arrive randomly and independently at a Doctor's room from 8 AM at an average rate of one in 5 minutes. The waiting room can accommodate 12 persons. The probability that the room will be full when the doctor arrives at 9AM is

  1. $\displaystyle \frac{{e}^{-12}(12)^{12}}{ 12!}$

  2. $\displaystyle \sum _{{x}=0}^{11}\frac{{e}^{-12}(12)^{{x}}}{ x!}$

  3. $1-\displaystyle \sum _{{x}=0}^{11}\frac{{e}^{-12}(12)^{{x}}}{ x!}$

  4. $1-\displaystyle \sum _{{x}=0}^{\infty }\frac{{e}^{-12}(12)^{{x}}}{ x!}$


Correct Option: C
Explanation:

Random Arrival process is a Poisson process
Given by
$P(k) = \dfrac{e^{-\lambda} \lambda^x}{x!}$
$given, \lambda = 1/5 min^{-1}= 12 s^{-1}$
Room is not full, if No.of patients $< 12$
Hence 
$P$(room is full) $= 1 - (P(1)+ . . . . +P(11))$
$=1-\displaystyle \sum _{{x}=0}^{11}\frac{{e}^{-12}(12)^{{x}}}{ x!}$

On an average, a submarine on patrol sights $6$ enemy ships per hour. Assuming the number of ships sighted in a given length of time is a Poisson variate, the probability of sighting at least two ships in the next $20$ minutes is

  1. $1-e^{-2}$

  2. $1-2e^{-2}$

  3. $1-3e^{-2}$

  4. $1-4e^{-2}$


Correct Option: C
Explanation:

The probability of sighting at least two ships
=1-(Probability of sighting atmost 1 ships)
$=1-e^{-\lambda}-\dfrac{\lambda^{1}e^{-\lambda}}{1!}$
Here $\lambda=2$
Hence 
$P=1-e^{-\lambda}-\dfrac{\lambda^{1}e^{-\lambda}}{1!}$
$=1-e^{-2}-2e^{-2}$
$=1-3e^{-2}$.

For a poisson distribution with parameter $\lambda = 0.25$, the value of the $2^{nd}$ moment about the origin is

  1. $0.25$

  2. $0.3125$

  3. $0.0625$

  4. $0.025$


Correct Option: B
Explanation:

Second moment about the origin is $\lambda +{ \lambda  }^{ 2 } = 0.25+{(0.25)}^{2} = 0.3125$
Therefore the correct option is $B$.

If $X$ is a Poisson's variate such that $P(X=1)=3P(X=2)$, then find the variance of $X$.

  1. $\cfrac 38$

  2. $\cfrac 13$

  3. $\cfrac 23$

  4. $\cfrac 54$


Correct Option: C
Explanation:
Fact:  Poisson distribution is $P(X=r)=\dfrac{e^{-\lambda}\lambda ^r}{r!}$

Now given $P(X=1)=3P(X=2)$

$\Rightarrow \dfrac{e^{-\lambda}\lambda}{1!}=3\cdot \dfrac{e^{-\lambda} \lambda^2}{2!}$

$\Rightarrow \lambda =\dfrac{2}{3}=$ Variance 

If X is a random poisson variate such that $E(X^2)=6$, then $E(x)=$?

  1. $3$

  2. $2$

  3. $-3$ & $2$

  4. $-2$


Correct Option: C
Explanation:

In poison distribution mean=variance=x

E(X)=x
variance=$E({ X }^{ 2 })$-${ E({ X }) }^{ 2 }$
$x$=6-${ x }^{ 2 }$
on solving we get x=-3 and 2

If $3 percent $ bulb manufactured by a company are defective; the probability that in a sample of $100$ bulbs exactly five defective is

  1. $\dfrac { { { e }^{ -0.003 } }\left( 0.03 \right) ^{ 5 } }{  5! }$

  2. $\dfrac { { { e }^{ -0.3 } }0.03^{ 5 } }{ 5!}$

  3. $\dfrac { { { e }^{ -3 } }3^{ 5 } }{5! }$

  4. $\dfrac { { e }^{ -0.3 }{ 3 }^{ -5 } }{5! }$


Correct Option: C
Explanation:

$In\quad poison\quad distribution\quad \frac { { e }^{ -u }{ u }^{ x } }{ x! } \ Here\quad 3\quad are\quad defective\quad in\quad 100\quad so\quad u=3\ Probaility\quad that\quad exactly\quad 5\quad are\quad defective\quad (x=5)=\quad \frac { { e }^{ -3 }{ (3) }^{ 5 } }{ 5! } $

If, in a Poisson distribution $P(X= 0)=k$ then the variance is: 

  1. $e^{\lambda}$

  2. $\log \dfrac{1}{k}$

  3. $\dfrac{1}{k}$

  4. $\log k$


Correct Option: B

The incidence of an occupational disease to the workers of a factory is found to be $\displaystyle \frac{1}{5000}$ . If there are $10,000$ workers in a factory then the probability that none of them will get the disease is

  1. $e^{-1}$

  2. $e^{-2}$

  3. $e^{3}$

  4. $e^{4}$


Correct Option: B
Explanation:
By using Poisson distribution, 
$ P(x;\mu)=\dfrac { { e }^{ -\mu }{ \mu }^{ x } }{ x! } $
$ \mu $: The mean number of successes that occur in a specified region.
$x: $ The actual number of successes that occur in a specified region.
$P(x;$$ \mu $ ): The Poisson probability that exactly x successes occur in a Poisson experiment, when the mean number of successes is $ \mu $
$ \mu = 10000 \times \dfrac{1}{5000} = 2 $
$x = 0$
$ P(0;2)=\dfrac { { e }^{ -2 }{ 2 }^{ 0 } }{ 0! } =  { e }^{ -2 }$

The probability that atmost $5$ defective fuses will be found in a box of $200$ fuses, if experience shows that $20 \%$ of such fuses are defective,  is

  1. $\displaystyle \frac{e^{-40}40^{5}}{ 5!}$

  2. $\displaystyle \sum _{x=0}^{5}\frac{e^{-40}40^{x}}{ x!}$

  3. $\displaystyle \sum _{x=6}^{\infty}\frac{e^{-40}40^{x}}{ x!}$

  4. $1-\displaystyle \sum _{x=6}^{\infty}\frac{e^{-40}40^{x}}{ x!}$


Correct Option: B
Explanation:

The poisson distribution is
$\displaystyle P(X=x)=\frac{e^{-\lambda }\lambda ^{x}}{x!}$ , $x=0,1,2,3,..$
Let, $X$ denote the defective fuse

$p=\frac{20}{100}$
$n=200$
mean$ = \lambda  = np = 200\times \frac{20}{100} = 40$
$\displaystyle => P($atmost $5$ defective fuses)$= _{x=0}^{5}\sum \frac{e^{-40}40^{x}}{x!}$

There are $500$ boxes each containing $1000$ ballot papers for election. The chance that a ballot paper is defective is $0.002$. Assuming that the number of defective ballot papers follow Poisson distribution, the number of boxes containing at least one defective ballot paper given that $e^{-2}=0.1353$ is

  1. $216$

  2. $432$

  3. $648$

  4. $234$


Correct Option: B
Explanation:

Here $\lambda = .002\times 1000=2$
Thus probability that a box contain at least one defective ballot $=1-P(X=0)=1-e^{-2}=1-0.1353=.8647$   
Hence   number of box out of $500$ which contain at least one defective ballot is $=.8647\times 500 =432$

Six unbiased coins are tossed $6400$ times. Using Poisson distribution, the approximate probability of getting six heads $2$ times is

  1. $\displaystyle \frac{e^{-64}(64)^{2}}{ 2!}$

  2. $\displaystyle \frac{e^{-100}(100)^{2}}{ 2!}$

  3. $1-\displaystyle \frac{e^{-100}(100)^{x}}{ x!}$

  4. $\displaystyle \frac{e^{-100}(100)^{x}}{x!}$


Correct Option: B
Explanation:

Here $\lambda = \left(\cfrac{1}{2}\right)^6\times 64100=100$
Hence probability of getting six heads two times $=P(X=2)=\cfrac{e^{-100}(100)^2}{2!}$

A company knows on the basis of past experience that $2$% of the blades are defective. The probability of having 3 defective blades in a sample of $100$ blades is

  1. $e^{-2}2^{2}$

  2. $\displaystyle \frac{e^{-2}2^{3}}{3!}$

  3. $\displaystyle \frac{e^{-2}2^{3}}{ 2!}$

  4. $\displaystyle \frac{e^{-4}2^{-1}}{ 2!}$


Correct Option: B
Explanation:

By using Poisson distribution, 
$ P(x;\mu)=\dfrac { { e }^{ -\mu }{ \mu }^{ x } }{ x! } $
$ \mu $: The mean number of successes that occur in a specified region(parameter)
x: The actual number of successes that occur in a specified region.
P(x; $ \mu $): The Poisson probability that exactly x successes occur in a Poisson experiment, when the mean number of successes is $ \mu $
Here,$ \mu  =  100 \times 0.02 = 2 $
$x = 3$(defective blades)
$ P(3;2)=\dfrac { { e }^{ -2 }{ 2 }^{3 } }{ 3! } $

A car hire firm has $2$ cars which it hires out day by day. If the number of demands for a car on each day follows poisson distribution with parameter $1.5$, then the probability that neither car is used is

  1. $e^{-1.5}$

  2. $1.5\times e^{-1.5}$

  3. $1-2.5\times e^{-1.5}$

  4. $1-1.5\times e^{-1.5}$


Correct Option: A
Explanation:

By using Poisson distribution, 
$ P(x;\mu)=\dfrac { { e }^{ -\mu }{ \mu }^{ x } }{ x! } $
$ \mu $: The mean number of successes that occur in a specified region(parameter)
x: The actual number of successes that occur in a specified region.
P(x; $ \mu $): The Poisson probability that exactly x successes occur in a Poisson experiment, when the mean number of successes is $ \mu $
Here, $ \mu  = 1.5$ and $x = 0$ (neither car is used)
$ P(0;1.5)=\dfrac { { e }^{ -1.5 }{ 1.5 }^{0 } }{ 0! } = {e}^{-1.5} $

In a big city, $5$ accidents take place over a period of $100$ days. If the numebr of accidents follows P.D., the probability that there will be $2$ accidents in a day is

  1. $\displaystyle \frac{e^{-5}5^{2}}{ 2!}$

  2. $\displaystyle \frac{e^{-05}5^{2}}{ 2!}$

  3. $\displaystyle \frac{e^{-005}(0.05)^{2}}{ 2!}$

  4. $\displaystyle \frac{e^{5}5^{2}}{ 2!}$


Correct Option: C
Explanation:
By using Poisson distribution, 
$ P(x;\mu)=\dfrac { { e }^{ -\mu }{ \mu }^{ x } }{ x! } $
$\mu\ :$ The mean number of successes that occur in a specified region.
$x\ :$ The actual number of successes that occur in a specified region.
$P(x;\mu )\ :$ The Poisson probability that exactly x successes occur in a Poisson experiment, when the mean number of successes is 
$ \mu = \dfrac {5}{100} = 0.05 $
$x = 2$ 
$ P(2; 005)=\dfrac { { e }^{ -0.05 }{ 0.05 }^{ 2 } }{ 2! } $

If ${ \mu  } _{ 2 }=20,{ \mu  } _{ 2 }^{ 1 }=276$ for a discrete random variable $X$, then the mean of the random variable $X$ is

  1. $16$

  2. $5$

  3. $2$

  4. $1$


Correct Option: A
Explanation:

$\mu _2=E(X^2)-[E(X)]^2......(i)$

$\mu _2'=E(X)^2.....(ii)$
Given: $\mu _2=20, \mu _2'=276$
from equation (ii) $\implies E(x^2)=276$
and from equation (i), $\implies 20=276-[E(X)]^2$
$\implies [E(X)]^2=276-20=250\\implies E(X)=16$
is the mean of random variable.

- Hide questions