Part CCSIR NET December 2024the-denominator-is-the-conditional-standard-deviation

The denominator is the conditional standard deviation

Let $(X_{1}, X_{2})$ be a bivariate normal random vector with $E (X_{1}) = 1, E (X_{2}) = 0, V a r (X_{1}) = 1, V a r (X_{2}) = 1$ , and correlation coefficient 1/2. Let U be a U(0, 1) random variable, which is independent of $(X_{1}, X_{2})$ . If Z = (UX $_{1} + X_{2} - U) / U^{2} + U + 1$ , then which of the following statements are true?

A.The distribution of Z is symmetric about 0.✓
B. $E (Z^{2}) = 2$
C. $V a r (Z^{2}) = 1$
D.Z and U are independent random variables.✓

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

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The denominator is the conditional standard deviation

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