Question

If $X_{1}, X_{2}, X_{3}$ are independent and identically distributed standard normal variates and let $U=\frac{\sqrt{2}X_{3}}{\sqrt{X_{1}^{2}+X_{2}^{2}}}$ then $U^{2}$ follows

• A. $t_{4}$
• B. $t_{2}$
• C. $F(1,2)$
• D. $F(1,4)$

Hint:

Remember that $t_n^2 = F(1, n)$. In this problem, $U$ itself actually follows a Student's $t$-distribution with 2 degrees of freedom ($t_2$), so its square naturally follows $F(1, 2)$.

Answer 1

The Correct Option is C