(Solved): 1) X is a Bernoulli random variable with Pr(X=1)=0.99;Y is distributed N(0,1);W is distributed N(0, ...

1) $X$ is a Bernoulli random variable with $\mathrm{Pr}(X=1)=0.99;Y$ is distributed $N(0,1);W$ is distributed $N(0,100)$; and $X,Y$, and $W$ are independent. Let $S=XY+(1?X)W$. (That is, $S=Y$ when $X=1$, and $S=W$ when $X=0$.) d. Derive $E(S),E\left(S^2\right),E\left(S^3\right)$, and $E\left(S^4\right)$. (Hint: Use the law of iterated expectations conditioning on $X=0$ and $X=1$.) e. Derive the skewness and kurtosis for $S$.