(Solved): Review of Conditional Expectation. Suppose Z is a random variable that is either 1 or 1 , with ...

Review of Conditional Expectation. Suppose $$Z$$ is a random variable that is either $$?1$$ or 1 , with probability $$0.5$$ of either outcome. If $$Z=?1$$, then $$X$$ is uniformly distributed between 0 and 4 . If $$Z=1$$, then $$X$$ is normally distributed with mean $$?2$$, and variance 4 . (a) Find $$\mathbb{E}[X]$$ (b) Find $$\mathrm{Var}(X)$$ (c) Verify parts (a) and (b) by drawing 10,000 samples from the distribution of $$X$$ and computing the mean and variance across these 10,000 samples.