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show that Z is standard normal

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.

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    $\begingroup$ For homework/self-study type questions please add the self-study tag $\endgroup$ – vigos Mar 5 at 10:44
  • $\begingroup$ 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. $\endgroup$ – Dilip Sarwate Mar 5 at 16:18
  • $\begingroup$ Previously asked: stats.stackexchange.com/questions/366806/…. $\endgroup$ – StubbornAtom Mar 10 at 6:36
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Hint:

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 $U_1,U_2$.

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