Discussion
Let \(X_{1}\) and \(X_{2}\) be two iid random variables with :
\(P(X_{1} = 1) = P(X_{1} = -1) = 1/2\).
Let \(Z = X_{1} + X_{2}\), \(A_{i} = X_{i}^{-1}(\{1\})\), \(i = 1, 2\).
Which of the following is not true on \(Z^{-1}(\{0\})\) ?
Answer: D
The statement that is not true is \(P(A_{1}|Z)P(A_{2}|Z) = P(A_{1} \cap A_{2}|Z)\).
On \(\{Z = 0\}\) the outcomes are (1, -1) and (-1, 1), each with conditional probability 1/2. \(A_{1}\) (that \(X_{1} = 1\)) happens in the first, so \(P(A_{1}|Z) = 1/2\), and likewise \(P(A_{2}|Z) = 1/2\). Both cannot happen, since then \(Z = 2\), so \(P(A_{1} \cap A_{2}|Z) = 0\).
The product of the conditional probabilities is \(1/4 \ne 0\), so the events, independent unconditionally, are not conditionally independent given \(Z = 0\). Options A, B and C are the true values.
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