Question Bank • QB_EM_ADV_071
MSQ • 2 Marks
Section 1: Engineering Mathematics
Probability and Statistics
Subtopic: Random variables
Question Statement:
Let $X$ and $Y$ be two independent random variables with finite expectations and variances. Which of the following statements is/are ALWAYS TRUE?
Options:
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(A) $E[X + Y] = E[X] + E[Y]$
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(B) $E[XY] = E[X] E[Y]$
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(C) $\operatorname{Var}(X + Y) = \operatorname{Var}(X) + \operatorname{Var}(Y)$
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(D) The covariance $\operatorname{Cov}(X, Y) = 0$
Step-by-Step Solution & Explanation
1. **Linearity of Expectation (Option A)**: Holds unconditionally for any random variables (True).
2. **Product Expectation under Independence (Option B)**: For independent variables, joint PDF factorizes: $E[XY] = \iint xy f_X(x) f_Y(y)\,dx\,dy = E[X]E[Y]$ (True).
3. **Variance of Sum (Option C)**: $\operatorname{Var}(X + Y) = \operatorname{Var}(X) + \operatorname{Var}(Y) + 2\operatorname{Cov}(X, Y)$. Under independence, $\operatorname{Cov}(X, Y) = 0$, so variances add directly (True).
4. **Covariance Zero (Option D)**: $\operatorname{Cov}(X, Y) = E[XY] - E[X]E[Y] = 0$ (True).
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