Question

Let $X_1$, $X_2$ be independent N(0,1) distributed random variables. Define $W_1 = X_1^2 + X_2^2$ and $W_2 = \frac{X_1^2 - X_2^2}{X_1^2 + X_2^2}$ . Without using calculus, show that $W_1 \perp W_2$.

Answer 1

without using calcutus, So we can't use P [ W, W2] = p (Wil. PlWe] I which is the actual method to prove independence. so only way here to prove is to use the properties properties of Independence as wello as the properties of Normal distribution DISKOGS DENON OCCA V PAD Y YUD W, = x ² + x 2 ~ X def. W = x² - x² x + x ² : W. of also holds the properties Normal distribution. Given that Xu Xy is independent NCOD)

then x + x and X - X are independent. - x²+x2 and x2 x ² are independent - (Bince linearity property of normal distsibution) is *,*+ independent of trantz x² + x₂ Thus we and we are independent