5. [9 points] (Mixture Distributions) Consider two continuous random variables Y and Z, and a random variable X that is equal to Y with probability p and Z with probability 1 p. (One way to think about this is that we ip a biased coin that comes up heads with probability p. The value of X is decided after rst ipping a coin if the coin comes up heads, then X = Y, otherwise X = Z) Show that (a) The PDF of X is given by fx(fr) = val?) + {1 - P)fz($)- (5-4) (b) Use the above expression to show that lE[X] plE [Y] + (l p)lE [Z]. Use the law of total probability to verify this. (c) Calculate the CDF of the two sided exponential random variable that has the PDF given by f ( ) pAeAx if it: