Question Bank • QB_EM_123
MSQ • 2 Marks
Section 1: Engineering Mathematics
Probability and Statistics
Subtopic: Random variables
Question Statement:
For any two independent random variables $X$ and $Y$, which of the following mathematical properties are ALWAYS satisfied?
Options:
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(A) $E(X Y) = E(X) \cdot E(Y)$
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(B) $\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y)$
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(C) $\text{Cov}(X, Y) = 0$
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(D) $\text{Var}(X - Y) = \text{Var}(X) - \text{Var}(Y)$
Step-by-Step Solution & Explanation
- A is true: Independence implies product expectation factorizes.
- B is true: $\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) + 2\text{Cov}(X, Y) = \text{Var}(X) + \text{Var}(Y)$.
- C is true: Covariance is identically zero for independent variables.
- D is false: $\text{Var}(X - Y) = \text{Var}(X) + \text{Var}(Y) - 2\text{Cov}(X, Y) = \text{Var}(X) + \text{Var}(Y)$, variances add.
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