Question

Let \( X \) and \( Y \) be independent random variables having \( \text{Bin}(18, 0.5) \) and \( \text{Bin}(20, 0.5) \) distributions, respectively. Further, let \( U = \min\{X, Y\} \) and \( V = \max\{X, Y\} \). Then which of the following statements is/are correct?

• A. \( E(U + V) = 19 \)
• B. \( E(|X - Y|) = E(V - U) \)
• C. \( \text{Var}(U + V) = 16 \)
• D. \( 38 - (X + Y) \) has \( \text{Bin}(38, 0.5) \) distribution

Answer 1

The Correct Option is A, B, D

1. Expectation of \( U + V \): - By definition, \( U + V = X + Y \), and: \[ E(X) = 18 \cdot 0.5 = 9, \quad E(Y) = 20 \cdot 0.5 = 10. \] Thus: \[ E(U + V) = E(X + Y) = E(X) + E(Y) = 9 + 10 = 19. \] 2. Expectation of \( |X - Y| \): - By the symmetry of \( X \) and \( Y \), \( E(|X - Y|) = E(V - U) \). 3. Variance of \( U + V \): - Since \( U + V = X + Y \), the variance is: \[ \text{Var}(U + V) = \text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) = 4.5 + 5 = 9.5, \] not 16. 4. Distribution of \( 38 - (X + Y) \): - Since \( X + Y \sim \text{Bin}(38, 0.5) \), \( 38 - (X + Y) \sim \text{Bin}(38, 0.5) \) by symmetry.