Problem 3: ISR5] Suppose you own a halloween store that sells masks. At the star

Question

Problem 3: ISR5] Suppose you own a halloween store that sells masks. At the start of the October, you order s masks. For each mask you sell in October, you earn b dollars For each mask left over at the end of the month, you lose dollars. Let X be a random variable of customers visiting your store who want to buy a mask, with probability mass function p(k)-P(X = k). Let Y be a randorn variable of your profit. (a) Define Y as a function of X. (b) Show that E[Y] sb + (b + 112:0(i-s)p(i) (c) Graduate students: find the optimal s to maximize your expected profit.

Explanation / Answer

Let x masks are sold in October.

a) Profit Y(x)  = bx l(sx)

  = (b + l)x ls

b) E[Y(x)] = [({b + l)i ls)}.p(i)] where i varies from 0 to s

  = [({b + l)i ls sb + sb)}.p(i)] where i varies from 0 to s

  = [({b + l)i s(l + b )+ sb)}.p(i)] where i varies from 0 to s

  = [({b + l)(i s)+ sb)}.p(i)] where i varies from 0 to s

= [({b + l)(i s)p(i)+ (sb). p(i)] where i varies from 0 to s

  = (b + l). (i s)p(i)+ sb where i varies from 0 to s ( p(i) is 1)

Hence Proved

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