# Deriving standard normal distribution from a statistic involving normal and uniform random variables I tried deriving distributions of numerator and denominator separately. But found that there is no closed form. I have no clue on how to show that Z is standard normal.

• For homework/self-study type questions please add the self-study tag – vigos Mar 5 at 10:44
• This must be a question in which the solver is expected to state which of the the four statements (marked 1 through 4 in the image) are true. Statement 3 is clearly incompatible with Statements 1, 2 and 4. Statement 4 is compatible with Statements 1 and 2 but need not be true in order for either 1 or 2 to be true. What exactly is the question actually asked? (Hint: this might be stated at the top of the page or at the beginning of the section. – Dilip Sarwate Mar 5 at 16:18
• Previously asked: stats.stackexchange.com/questions/366806/…. – StubbornAtom Mar 10 at 6:36

Consider the conditional distribution $$Z\mid( U_1=u_1,U_2=u_2)$$ and use what you know about the reproductive property of univariate normal distribution. You would find that $$Z\mid U_1,U_2$$ is independent of $$U_1,U_2$$.