(A) The Poisson distribution is given by the formula $P(X=r) = \frac{e^{-m} m^r}{r!}$,where $m$ is the mean number of occurrences.Given $m = 5$,we wan

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(A) The Poisson distribution is given by the formula $P(X=r) = \frac{e^{-m} m^r}{r!}$,where $m$ is the mean number of occurrences.Given $m = 5$,we want to find the probability of at most one phone call,which is $P(X \leq 1) = P(X=0) + P(X=1)$.For $r=0$: $P(X=0) = \frac{e^{-5} 5^0}{0!} = e^{-5}$.For $r=1$: $P(X=1) = \frac{e^{-5} 5^1}{1!} = 5e^{-5}$.Adding these probabilities: $P(X \leq 1) = e^{-5} + 5e^{-5} = 6e^{-5}$.

Class 12 Mathematics Probability Probability distribution

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

AIEEE 2006 All AIEEE

A
$6e^{-5}$
B
$5e^{-5}$
C
$e^{-5}$
D
$4e^{-5}$

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The cumulative distribution function (c.d.f.) $F(x)$ of a discrete random variable $X$ is given by the following table:
$X$ $-3$ $-1$ $0$ $1$ $3$ $5$ $7$ $9$
$F(X=x)$ $0.1$ $0.3$ $0.5$ $0.65$ $0.75$ $0.85$ $0.90$ $1$

Then,find the value of $\frac{P[X=-3]}{P[X < 0]}$.

$X$	$-3$	$-1$	$0$	$1$	$3$	$5$	$7$	$9$
$F(X=x)$	$0.1$	$0.3$	$0.5$	$0.65$	$0.75$	$0.85$	$0.90$	$1$

EasyMHT CET 2025

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Let $X$ be the random variable taking values $1, 2, \ldots, n$ for a fixed positive integer $n$. If $P(X=k) = \frac{1}{n}$ for $1 \leq k \leq n$,then the variance of $X$ is

MediumAP EAMCET 2025

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$A$ random variable $X$ has the following probability distribution:
$X$ $1, 2, 3, 4, 5$
$P(X)$ $K^2, 2K, K, 2K, 5K^2$

Then $P(X > 2)$ is equal to:

DifficultMHT CET 2024

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Three rotten apples are mixed accidentally with seven good apples and four apples are drawn one by one without replacement. Let the random variable $X$ denote the number of rotten apples. If $\mu$ and $\sigma^2$ represent the mean and variance of $X$,respectively,then $10(\mu^2 + \sigma^2)$ is equal to

DifficultJEE MAIN 2023

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The variance of a Poisson variate $X$ is $2$. Then $P(X \geq 3) = $

EasyTS EAMCET 2022

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

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The cumulative distribution function (c.d.f.) $F(x)$ of a discrete random variable $X$ is given by the following table:$X$$-3$$-1$$0$$1$$3$$5$$7$$9$$F(X=x)$$0.1$$0.3$$0.5$$0.65$$0.75$$0.85$$0.90$$1$Then,find the value of $\frac{P[X=-3]}{P[X < 0]}$.

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The cumulative distribution function (c.d.f.) $F(x)$ of a discrete random variable $X$ is given by the following table:
$X$ $-3$ $-1$ $0$ $1$ $3$ $5$ $7$ $9$
$F(X=x)$ $0.1$ $0.3$ $0.5$ $0.65$ $0.75$ $0.85$ $0.90$ $1$

Then,find the value of $\frac{P[X=-3]}{P[X < 0]}$.

$A$ random variable $X$ has the following probability distribution:
$X$ $1, 2, 3, 4, 5$
$P(X)$ $K^2, 2K, K, 2K, 5K^2$

Then $P(X > 2)$ is equal to: