(Solved): 7. (4 points) Let X, Y and Z be three mutually independent random variables following the distri- b ...

7. (4 points) Let X, Y and Z be three mutually independent random variables following the distri- butions X~ Bernoulli (0.99), Define a new variable, S, by S= Y~N(1,1), Z~ N(-10, 100). Y if X = 1 Z if X=0. To find the expectation of S, someone proposes the following method: E[S]E[SIX = 1] x P[X = 1] + E[S|X=0] x P[X = 0] E[Y] x P[X = 1] + E[Z] x P[X = 0]. (a) In words, explain the reasoning behind the two equalities labeled by (i) and (ii). (3 points) (b) Find E[S]. (1 point)