91Ó°ÊÓ

To find the conditional expectation, we can use the following formula: \[E[X \mid X \neq 0] = \frac{E[X \cdot 1_{\{X \neq 0\}}]}{P(X \neq 0)}\] First, we need to find the probability that X is not equal to 0, which can be calculated using the complement rule, so we have \(P(X \neq 0) = 1 - P(X=0) = 1 - p_0\). Next, we need to find the expected value of X times the indicator function, which is: \[E[X \cdot 1_{\{X \neq 0\}}] = E[X] - E[X \cdot 1_{\{X = 0\}}]\] Since the indicator function is 1 when X is not equal to 0 and 0 otherwise, we have: \[ E[X \cdot 1_{\{X \neq 0\}}] = E[X] - E[X \cdot 1_{\{X = 0\}}] = E[X] - 0 * P(X=0) = E[X] = \mu \] Now, we can calculate the conditional expectation: \[ E[X \mid X \neq 0] = \frac{E[X \cdot 1_{\{X \neq 0\}}]}{P(X \neq 0)} = \frac{\mu}{1 - p_0} \]

We can find the conditional variance of X given that X is not equal to 0 using the following formula: \[ \operatorname{Var}(X \mid X \neq 0) = E[X^2 \mid X \neq 0] - \left(E[X \mid X \neq 0]\right)^2 \] First, let's find the conditional expectation of \(X^2\), similar to how we found the conditional expectation of X in part (a): \[ E[X^2 \mid X \neq 0] = \frac{E[X^2 \cdot 1_{\{X \neq 0\}}]}{P(X \neq 0)} = \frac{E[X^2] - E[X^2 \cdot 1_{\{X = 0\}}]}{1 - p_0} \] Since \(E[X^2 \cdot 1_{\{X = 0\}}] = 0 * P(X=0) = 0\), we have: \[ E[X^2 \mid X \neq 0] = \frac{E[X^2]}{1 - p_0} \] Using the variance of X, we know that \(\sigma^2 = E[X^2] - \mu^2\), so we can rewrite the conditional expectation of \(X^2\) as: \[ E[X^2 \mid X \neq 0] = \frac{\sigma^2 + \mu^2}{1 - p_0} \] Now, we can calculate the conditional variance: \[ \operatorname{Var}(X \mid X \neq 0) = E[X^2 \mid X \neq 0] - \left(E[X \mid X \neq 0]\right)^2 = \frac{\sigma^2 + \mu^2}{1 - p_0} - \left(\frac{\mu}{1 - p_0}\right)^2 \] To make it look more appealing, we may rewrite it as: \[ \operatorname{Var}(X \mid X \neq 0) = \frac{\sigma^2(1 - p_0) + \mu^2 - \mu^2}{(1 - p_0)^2} = \frac{\sigma^2(1 - p_0)}{(1 - p_0)^2} \] So, the conditional variance of X given that X is not equal to 0 is: \[ \operatorname{Var}(X \mid X \neq 0) = \frac{\sigma^2(1 - p_0)}{(1 - p_0)^2} \]