1

Find the distribution of $W= \frac{X_1+X_2}{|X_1-X_2|}$ , $X_i = N(0,\sigma^2).$
Moreover: What hypothesis testing can be performed using the statistic in Q1 when is unknown?

My attempt:
I know, by using moment generating functions and using any stats textbook that $X_1+X_2$ has the distribution $N(0,2\sigma^2)$.

However this is where I am stuck.

I know that $X_1-X_2$ has the distribution $N(0,2\sigma^2)$ however I am unsure whether the absolute value makes a difference. If it doesn't the Wolfram Alpha page on Cauchy Distributions says that the ratio of two random normal variables centered at 0 is Cauchy. However I am not sure whether this is the case, and if it is the case, what assumptions can be made about the hypothesis testing. Any clarification would be great.

Matthew
  • 1,354
  • Question is incomplete. If $X_1,X_2$ are independent, then $W$ is definitely standard Cauchy. See https://math.stackexchange.com/questions/2161566/find-the-distribution-of-frac-xy-given-that-x-and-y-are-independent?noredirect=1&lq=1, https://math.stackexchange.com/questions/948527/distribution-of-fracxy-where-x-and-y-are-standard-normal-r-v-s?noredirect=1&lq=1. – StubbornAtom Apr 09 '19 at 18:47

0 Answers0