X and Y are independent standard normal random variables.
Prove following are also a standard normal random variables.
If they are not explain why.
1. (X+Y) /2
2. What is the mean? Therefore ....
3. Use Google to discover why U = X/Y is a Cauchy random variable with shape factor 1 and median zero. Or calculate the pdf in one of several ways eg. calculate and then differentiate the cdf, or use a well known theorem given in 'The Algebra of Random Variables' by M.D. Springer (or find and then use this theorem using Google: Key words 'ratio distribution' or 'quotient random variables' or other such key words).
4. What is the variance? Therefore ....
5. The pdf of the sum or difference of two random variables can be found in several ways eg. convolution, calculating and then differentiating the cdf, use of moment generating functions etc. I suggest you use the method you have been taught to show that U = (X - Y)/2 is a standard normal variable.