Question

Let \( X_1, X_2, \ldots, X_{20} \) be a random sample from the \( N(5, 2) \) distribution, and let \( Y_i = X_{2i} - X_{2i-1} \) for \( i = 1, 2, \ldots, 10 \). Then \( W = \frac{1}{4} \sum_{i=1}^{10} Y_i^2 \) has the distribution:

• A. \( t_{20} \) distribution
• B. \( \chi^2_{20} \) distribution
• C. \( \chi^2_{10} \) distribution
• D. \( N(250, 20) \) distribution

Answer 1

The Correct Option is C

1. Distribution of \( Y_i \): - Since \( X_i \sim N(5, 2) \), the difference \( Y_i = X_{2i} - X_{2i-1} \) follows: \[ Y_i \sim N(0, 4). \] 2. Distribution of \( W \): - The sum of squared \( Y_i \) values scaled by \( \frac{1}{4} \) gives: \[ W = \frac{1}{4} \sum_{i=1}^{10} \frac{Y_i^2}{4}. \] - This follows a \( \chi^2_{10} \) distribution, as it is a sum of squares of 10 independent standard normal random variables.