Question

Let \( X_1, X_2, \dots, X_m, Y_1, Y_2, \dots, Y_n \) be i.i.d. \( N(0,1) \) random variables. Then \[ W = \frac{n \sum_{i=1}^m X_i^2}{m \sum_{j=1}^n Y_j^2} \] has

• A. \( X_{n+m}^2 \) distribution
• B. \( t_n \) distribution
• C. \( F_{m,n} \) distribution
• D. \( F_{1,n} \) distribution

Hint:

When working with sums of squared normal variables, the resulting distribution is often an \( F \)-distribution.

Answer 1

The Correct Option is D

Step 1: Understanding the problem.
The random variables \( X_1, X_2, \dots, X_m, Y_1, Y_2, \dots, Y_n \) are i.i.d. standard normal random variables. The expression for \( W \) involves a ratio of sums of squared normal variables.
Step 2: Identifying the distribution.
The distribution of \( W \) follows an \( F \)-distribution. Specifically, it is a ratio of two scaled sums of squared normal variables, and since there are one set of squared variables from \( X \) and another from \( Y \), \( W \) follows the \( F \)-distribution with 1 degree of freedom in the numerator and \( n \) degrees of freedom in the denominator, denoted by \( F_{1,n} \).
Step 3: Conclusion.
Thus, \( W \) has an \( F_{1,n} \) distribution, and the correct answer is (D).