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To find the conditional expectation \(E[X \mid X \neq 0]\), we need to consider what the expected value of \(X\) would be if we ignored the cases where \(X = 0\). We can calculate this by using the definition of conditional expectation: \[E[X \mid X \neq 0] = \frac{E[X \cdot 1_{\{ X \neq 0\}}]}{P[X \neq 0]}\] where \(1_{\{ X \neq 0\}}\) is the indicator function that takes on the value 1 when \(X \neq 0\) and 0 otherwise. Notice that when \(X = 0\), the product \(X \cdot 1_{\{ X \neq 0\}}\) is also 0. Therefore, we can rewrite the conditional expectation as a ratio that only considers the non-zero values of \(X\): \[E[X \mid X \neq 0] = \frac{E[X]}{P[X \neq 0]}\] Since we know \(\mu = E[X]\) and \(p_0 = P[X = 0]\), we can easily find \(P[X \neq 0]\): \[P[X \neq 0] = 1 - P[X = 0] = 1 - p_0\] Finally, we can plug in our known values of \(\mu\) and \(1 - p_0\) to calculate the conditional expectation: \[E[X \mid X \neq 0] = \frac{\mu}{1 - p_0}\]

Now let's find the conditional variance \(\operatorname{Var}(X \mid X \neq 0)\). We can make use of the property that relates conditional variance to conditional second moments: \[\operatorname{Var}(X \mid X \neq 0) = E[X^2 \mid X \neq 0] - (E[X \mid X \neq 0])^2\] We have already calculated \(E[X \mid X \neq 0]\) in part (a). What remains is to find \(E[X^2 \mid X \neq 0]\). Just like in the conditional expectation calculation, we can express the conditional second moment as a ratio involving the non-zero values of \(X\): \[E[X^2 \mid X \neq 0] = \frac{E[X^2 \cdot 1_{\{ X \neq 0\}}]}{P[X \neq 0]}\] We have already found \(P[X \neq 0]\) in part (a). Now we need to find the unconditional second moment \(E[X^2 \cdot 1_{\{ X \neq 0\}}]\). Notice that: \[E[X^2] = P[X=0]\cdot(0)^2 + E[X^2 \mid X \neq 0] \cdot P[X \neq 0]\] We have all the information we need to calculate \(E[X^2]\). Using the formula \(\sigma^2 = \operatorname{Var}(X) = E[X^2] - E[X]^2\), we can solve for \(E[X^2]\): \[E[X^2] = \sigma^2 + \mu^2\] Now we can plug in the values for \(E[X^2]\) and \(P[X \neq 0]\) to find \(E[X^2 \mid X \neq 0]\): \[E[X^2 \mid X \neq 0] = \frac{\sigma^2 + \mu^2}{1 - p_0}\] Finally, we can use our results for \(E[X \mid X \neq 0]\) and \(E[X^2 \mid X \neq 0]\) in the conditional variance formula: \[\operatorname{Var}(X \mid X \neq 0) = \left(\frac{\sigma^2 + \mu^2}{1 - p_0}\right) - \left(\frac{\mu}{1 - p_0}\right)^2\]