Example Variance of a sum of a random number of random variables 5q Let X1, X2,. . . be a sequence of independent and identically distributed random vari- ables, and let / be a nonnegative integer-valued random variable that is independent N of the sequence X,i 2 1. To compute Var ( E X , we condition on N: N E EXiN = NE[X] i=1 N Var XIN = N Var(X) i=1 The preceding result follows because, given N. E X, is just the sum of a fixed number of independent random variables, so its expectation and variance are just the sums of the individual means and variances, respectively. Hence, from the condi- tional variance formula, N Var EXi = E[N]Var(X) + (E[X])-Var(N) =1A store owner decided to use a promotion to decrease the supply level of a certain overstocked item [item A] for one day. The marked price of the promotion item is $200. For each customerI the store owner will roll a fair die, and apply a ll'ii: discount on the item A where I is the outcome of a dice roll. For example, if I = 3, the sale price of the item A is E 0.3} = $14. The number of customers who visit the store during the promotion day follows a Poisson distribution with parameter 2i]. Let X he the dollar amount of sales from the promotion item on the promotion day. {a} 1|What is the average selling price of the item A on the promotion day? {h} Suppose all customers who visited the store on the promotion day bought the item A. Find E[X] and iv'ar{X} [c] Suppose the store also sells a similar item {item B} with the marl-ted price of $150. Suppose every customer who visits the store buys only one of item A or item B? whichever is lEEE expensive. Let 2'! he the total amount of sales from the two items [item A and E}. Find spa] and VarIEZ)