Algebra

Probability

Medium

applied

If X is a random variable and a, b are real numbers, then which of the following statements are correct?

(A) $E[aX+b] = a E(X) + b$

(B) $Var (aX + b) = a^2 Var (X) + b$

(C) $Var (aX + b) = a Var (X)$

(D) $Var (X) = E(X^2) - [E(X)]^2$

Choose the correct answer from the options given below:

(A), (B) and (D) only

(A) and (D) only

(B) and (C) only

(A), (C) and (D) only

✅ Correct Option: 2

For statement (A): $E[aX+b] = a E(X) + b$

The linearity of expectation gives us that multiplying a random variable by constant $a$ multiplies the expectation by $a$ , and adding constant $b$ adds to the expectation.

Statement (A) is correct.

For statement (B): $Var(aX + b) = a^2 Var(X) + b$

The correct formula is $Var(aX + b) = a^2 Var(X)$ .

Adding a constant $b$ does not change variance since variance measures spread. Shifting all values by constant $b$ does not change the spread.

Statement (B) is incorrect.

For statement (C): $Var(aX + b) = a Var(X)$

The correct formula is $Var(aX + b) = a^2 Var(X)$ .

Multiplying by constant $a$ multiplies variance by $a^2$ , not $a$ .

Statement (C) is incorrect.

For statement (D): $Var(X) = E(X^2) - [E(X)]^2$

This is the standard variance formula where:

$E(X^2)$ is the average of squares

$[E(X)]^2$ is the square of average

Variance equals the average of squares minus the square of average.

Statement (D) is correct.

Only statements (A) and (D) are correct.

The correct answer is Option 2: (A) and (D) only.

2025: 2 June Shift 1

An urn contains 5 red and 5 black balls. A ball is drawn at random, its color is noted and is returned to the urn. Moreover, 2 additional balls of the same color are put in the urn and then a ball is drawn at random. The probability that the second drawn ball is red, is:

2025: 3 June Shift 1

Which of the following statements are correct?

(A) The mean and variance of the Poisson distribution are equal.

(B) The mean and variance of a Binomial distribution are equal.

(C) An unbiased die is thrown again and again until two sixes are obtained, then the probability of obtaining the second six in the 3rd throw is $\frac{5}{108}$ .

(D) If the variance of a Poisson distribution is 2, then P(X = 2) = $2e^{-2}$

Choose the correct answer from the options given below:

2025: 22 May Shift 2

Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. The probability distribution of number of aces is given by:

2026: 18 May Shift 1

If A and B are two events such that $P(A) \neq 0$ , $P(B) \neq 1$ and $P(\bar{A}/\bar{B}) = \frac{1-k}{P(\bar{B})}$ , then the value of $k$ is

2023: 15 June Shift 2

A and B throw a die alternatively till one of them gets a number more than 4 and wins the game. Then the probability of winning the game by B, if A starts first :

2025: 3 June Shift 2

A bag contains 4 red and 6 black balls. Two balls are drawn in succession without replacement. The probability that the first is red and the second is black is

Solution

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