Applied Mathematics

Probability Distribution

Medium

applied

The mean and variance of a random variable $X$ having a binomial distribution are 1 and 2/3 respectively. Then,

Match the LIST-I with LIST-II

LIST-I LIST-II
A. $P(X=0)$ I. $1/27$
B. $P(X=3)$ II. $2/9$
C. $P(X=2)$ III. $4/9$
D. $P(X=1)$ IV. $8/27$

Choose the correct answer from the options given below:

LIST-I	LIST-II
A. $P(X=0)$	I. $1/27$
B. $P(X=3)$	II. $2/9$
C. $P(X=2)$	III. $4/9$
D. $P(X=1)$	IV. $8/27$

A-I, B-IV, C-II, D-III

A-I, B-IV, C-III, D-II

A-IV, B-I, C-III, D-II

A-IV, B-I, C-II, D-III

✅ Correct Option: 4

$np=1$ and $npq=\dfrac23$ give $q=\dfrac23$ , $p=\dfrac13$ , $n=3$ .

$P(0)=\left(\frac23\right)^3=\dfrac{8}{27}$ (IV), $P(3)=\dfrac{1}{27}$ (I).

$P(2)=3\cdot\dfrac19\cdot\dfrac23=\dfrac29$ (II), $P(1)=3\cdot\dfrac13\cdot\dfrac49=\dfrac49$ (III).

2023: 23 May Shift 3

The random variable X has a probability distribution P(X) of the following form where k is a scalar and

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

then value of P(X < 2) = _______.

2024: 16 May Shift 1

$X$	-2	-1	0	1	2
$P(X)$	$0.2$	$0.1$	$0.3$	$0.2$	$0.2$

The variance of $X$ will be :

2023: 23 May Shift 3

A book consisting of 2000 pages has 540 misprints distributed randomly throughout the book. The average number of misprints in one page of the book is:

2026: 14 May Shift 2

If X has a Poisson distribution such that $2P(X=1)=P(X=2)$ , then which of the following are correct?

A. The mean of X is 4

B. The mean of X is 2

C. $P(X=2)=\frac{8}{e^4}$

D. $P(X=3)=\frac{32}{e^4}$

Choose the correct answer from the options given below:

2022: 25 May Shift 1

Match List I with List II

LIST I	LIST II
A. The variance of a Poisson distribution with mean $\lambda$ is	I. $\sqrt{\lambda}$
B. The standard deviation of a Poisson distribution with mean $\lambda$ is	II. 4
C. In a Poisson distribution, if mean is 4, then the standard deviation is	III. $\lambda$
D. In a Poisson distribution, if mean is 4, then the variance is	IV. 2

Choose the correct answer from the options given below:

2026: 26 May Shift 2

In a random experiment, a collection of trials is called Bernoulli trials then which of the following statements are TRUE?

A. The number of trials is infinite.

B. Each trial has exactly two outcomes defined as success and failure.

C. The trials are dependent by nature.

D. The probability of success remains the same in each trial.

Choose the correct answer from the options given below:

Solution

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