If X is a discrete random variable which assumes values x_1, x_2, x_3, ..., x_n with the respective probabilities p_1, p_2, ....., p_n, then Match the LIST-I with LIST-II, LIST-I, LIST-II, A., _(i=1)^n p_i =, I., E(X^2), B., _(i=1)^n p_i x_i =, II., Var(X), C., _(i=1)^n p_i x_i^2 =, III., 1, D., E(X^2) - \E(X)\^2 =, IV., E(X), Choose the correct answer from the options given below: Option 1: A-I, B-IV, C-II, D-III Option 2: A-IV, B-I, C-III, D-II Option 3: A-II, B-I, C-IV, D-III Option 4: A-III, B-IV, C-I, D-II

Answer: A-III, B-IV, C-I, D-II. Solution: Probabilities of a distribution sum to 1 (III). The weighted sum p_ix_i is the mean E(X) (IV). The sum p_ix_i^2 is E(X^2) (I). Finally E(X^2)-\E(X)\^2 is the variance (II).

Applied Mathematics

Probability Distribution

Easy

applied

If $X$ is a discrete random variable which assumes values $x_1, x_2, x_3, ..., x_n$ with the respective probabilities $p_1, p_2, ....., p_n$ , then

Match the LIST-I with LIST-II

LIST-I LIST-II
A. $\displaystyle\sum_{i=1}^{n} p_i =$ I. $E(X^2)$
B. $\displaystyle\sum_{i=1}^{n} p_i x_i =$ II. $Var(X)$
C. $\displaystyle\sum_{i=1}^{n} p_i x_i^2 =$ III. $1$
D. $E(X^2) - \{E(X)\}^2 =$ IV. $E(X)$

Choose the correct answer from the options given below:

	LIST-I		LIST-II
A.	$\displaystyle\sum_{i=1}^{n} p_i =$	I.	$E(X^2)$
B.	$\displaystyle\sum_{i=1}^{n} p_i x_i =$	II.	$Var(X)$
C.	$\displaystyle\sum_{i=1}^{n} p_i x_i^2 =$	III.	$1$
D.	$E(X^2) - \{E(X)\}^2 =$	IV.	$E(X)$

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

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

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

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

✅ Correct Option: 4

Probabilities of a distribution sum to 1 (III). The weighted sum $\sum p_ix_i$ is the mean $E(X)$ (IV). The sum $\sum p_ix_i^2$ is $E(X^2)$ (I). Finally $E(X^2)-\{E(X)\}^2$ is the variance (II).

2026: 29 May Shift 2

If 0.1% of the bolts manufactured by a machine are found to be defective. Find the probability that, in a sample of 2000 bolts chosen at random, exactly 3 will be defective using the Poisson distribution. [Given $e^{-2} =0.1353$ ]

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

2026: 12 May Shift 2

Ajay is a vendor who informs that there are 2% rotten apples in the fruit bag. Find the probability that a sample of 10 apples will include not more than one rotten apple.

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:

2026: 21 May Shift 2

If a random variable X follows binomial distribution with mean $\dfrac{4}{3}$ and variance $\dfrac{8}{9}$ . Which of the following are correct?

A. Probability of success is $\dfrac{1}{3}$ .

B. Probability of failure is $\dfrac{2}{3}$ .

C. Number of trials is 4.

D. $P(X \ge 1) = \dfrac{35}{81}$

Choose the correct answer from the options given below:

2023: 22 May Shift 3

If the probability distribution of X is :

X	2	3	4	5	6
P(X)	1/15	2/15	3/15	4/15	5/15

Then variance is equal to :

Solution

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