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. \( F_{1,n} \) distribution
• D. \( F_{m,n} \) distribution

Hint:

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

Answer 1

The Correct Option is C

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 \( F_{m, n} \), because it is the ratio of two scaled sums of squared normal variables (which follows the definition of an \( F \)-distribution).
Step 3: Conclusion.
Thus, \( W \) has an \( F_{m,n} \) distribution, and the correct answer is (C).

Answer 2

Step 1: Identify the distribution of \( X_i \) and \( Y_j \).
We are given that \( X_1, X_2, \dots, X_m, Y_1, Y_2, \dots, Y_n \) are i.i.d. random variables, each distributed as \( N(0,1) \) (standard normal distribution).
This means each \( X_i \) and \( Y_j \) follows the standard normal distribution with mean 0 and variance 1.

Step 2: Understand the distribution of the sum of squares of normal random variables.
For a random variable \( Z \sim N(0,1) \), the square of \( Z \), i.e., \( Z^2 \), follows a chi-square distribution with 1 degree of freedom, denoted as \( \chi^2_1 \).
Thus, each \( \sum_{i=1}^m X_i^2 \) is the sum of \( m \) independent chi-square distributed random variables, each with 1 degree of freedom, i.e., it follows \( \chi^2_m \). Similarly, \( \sum_{j=1}^n Y_j^2 \) follows \( \chi^2_n \).

Step 3: Analyze the ratio.
We are interested in the random variable: \[ W = \frac{n \sum_{i=1}^m X_i^2}{m \sum_{j=1}^n Y_j^2} \] The numerator \( n \sum_{i=1}^m X_i^2 \) follows \( n \times \chi^2_m \), and the denominator \( m \sum_{j=1}^n Y_j^2 \) follows \( m \times \chi^2_n \).
The ratio of two independent chi-square random variables (scaled by their degrees of freedom) follows an \( F \)-distribution.
Thus, \( W \) follows an \( F \)-distribution with degrees of freedom \( m \) and \( n \), i.e., \( W \sim F_{m,n} \).

Step 4: Final Answer.
The distribution of \( W \) is \( F_{m,n} \). Since we are given \( m = 1 \), the distribution is \( F_{1,n} \).

Final Answer: \( F_{1,n} \) distribution