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

The denominator is the conditional standard deviation

Let (X1,X2)(X_{1}, X_{2}) be a bivariate normal random vector with E(X1)=1,E(X2)=0,Var(X1)=1,Var(X2)=1E(X_{1}) = 1, E(X_{2}) = 0, Var(X_{1}) = 1, Var(X_{2}) = 1, and correlation coefficient 1/2. Let U be a U(0, 1) random variable, which is independent of (X1,X2)(X_{1}, X_{2}). If Z = (UX1+X2−U)/U2+U+1_{1} + X_{2} - U)/\sqrt{U^{2} + U + 1}, then which of the following statements are true?

  1. A.The distribution of Z is symmetric about 0.✓ Correct
  2. B.E(Z2)=2E(Z^{2}) = 2
  3. C.Var(Z2)=1Var(Z^{2}) = 1
  4. D.Z and U are independent random variables.✓ Correct

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, and why this one tests dependence misread.

See pricing

50 are analysed free — try those first.

The trap it tests

Dependence misread

Independence, pairwise vs mutual, or correlation vs dependence.

Drill statements like this

Related counterexample: If every marginal is normal then the vector is multivariate normal

More on this topic

The chapter behind this: Multivariate normal and Wishart — free to read

From Linear Models and Multivariate › Multivariate normal distribution

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

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