Applied Mathematics

Probability Distribution

Medium

applied

A telephone exchange receives on an average 5 calls per minute. The probability of receiving 3 or less calls per minute is :

$15e^{-5}$

$\frac{118}{3}e^{-5}$

$\frac{15}{4}e^{-5}$

$\frac{13}{3}e^{-4}$

✅ Correct Option: 2

Poisson with $\lambda = 5$ :

$P(X \leq 3) = e^{-5}\left[1 + 5 + \frac{25}{2} + \frac{125}{6}\right]$

$= e^{-5} \cdot \frac{6 + 30 + 75 + 125}{6} = \frac{236}{6}e^{-5} = \frac{118}{3}e^{-5}$ .

2025: 16 May Shift 1

If the random variable X follows the Poisson distribution such that P[X = k] = P[X = k+1], then the mean value of X is:

2023: 11 Aug Shift 1

$P(X = x) = \begin{cases} 2k & \text{if } x = 0 \\ kx & \text{if } x = 1 \\ k(x - 1) & \text{if } x = 2 \text{ or } 3 \\ 0 & \text{otherwise} \end{cases}$

The value of k is

2023: 23 May Shift 3

If the mean of a binomial distribution is 12 and its standard deviation is 2, then the number of trials is :

2026: 25 May Shift 1

If $X$ is a random variable such that $X$ can take values 0,1, 2 or 3. Then the expectation of $X$ for the following data is :

$X$	0	1	2	3
$P(X)$	$k$	$k^2$	$1-5k^2$	$k^2$

(where $k>0$ )

2026: 14 May Shift 2

Match the LIST-I with LIST-II

	LIST-I		LIST-II
A.	In a binomial distribution, if $n = 16$ , $p = 0.75$ , then the mean is	I.	$4$
B.	In a binomial distribution, if $n = 25$ , $p = \dfrac{1}{5}$ , then variance is	II.	$32$
C.	In a binomial distribution, the mean is 8 and the variance is 6, then the number of trials is	III.	$18$
D.	If the mean and variance of a binomial distribution are 6 and 4 respectively, then the number of trials is	IV.	$12$

Choose the correct answer from the options given below:

2023: 11 Aug Shift 1

The probability distribution of a discrete random variable X is given by :

X	30	10	-10
P(X)	1/5	3/10	1/2

then E(X) is equals to

A telephone exchange receives on an average 5 calls per minute. The probability of receiving 3 or less calls per minute is :

Solution

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