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?
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?
Updated On: Jan 25, 2025
- \( E(U + V) = 19 \)
- \( E(|X - Y|) = E(V - U) \)
- \( \text{Var}(U + V) = 16 \)
- \( 38 - (X + Y) \) has \( \text{Bin}(38, 0.5) \) distribution
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The Correct Option is A, B, D
Solution and Explanation
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.
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