1. A small drugstore orders copies of a certain magazine to display each week. L
ID: 2933021 • Letter: 1
Question
1. A small drugstore orders copies of a certain magazine to display each week. Let X be the weekly demand for that magazine with PMF x=1, 2, 3, 4 TS f(x)-- x=5,6 15 0 otherwise (a) Compute E[X] and Var[X]. (b) Suppose the store owner actually pays $1 to buy each copy of the magazine and sells to customers for $2 per copy. If magazines left at the end of the week have no salvage value and any unmet demand is lost, then is it better to stock 3 or 4 copies of the magazine at the start ofa week to maximize the expected profit?Explanation / Answer
(a) f(1) = 1/1
f(2) = 2/15
f(3) = 3/15
f(4) = 4/15
f(5) = 3/15
f(6) = 2/15
E(X) = 1/15 [ 1 * 1 + 2 * 2 + 3 * 3 + 4 * 4 + 5 * 3 + 6 *2] = 3.8
Var(X) = E(X2) - E(X)2
E(X2) = 1/15[ 12* 1 + 22 * 2 + 32 * 3 + 42 * 4 + 52 * 3 + 62 * 2] = 247/15
Var(X) = 247/15 - 3.82 = 2.0267
(b) Let check stock level 2 to 5
he stocks 2 copies of the megazine
Total Expected earnings= Pr(1 sell) * 2 + Pr(2 sell) * 4 = 1/15 [1 * 2 + 14 * 4 ] = $ 3.87
Total expected cost = 2* 1 = $ 2
Profit = $ 3.87 - $ 2= $ 1.87
he stocks 3 copies of the megazine
Total Expected earnings= Pr(1 sell) * 2 + Pr(2 sell) * 4 + Pr(3 sell) * 6 = 1/15 [1 * 2 + 2 * 4 + 12 * 6] = $ 5.47
Total expected cost = 3 * 1 = $ 3
Profit = $ 5.47 - $ 3 = $ 2.47
Let say he stocks 4 copies of the megazine
Total Expected earnings= Pr(1 sell) * 2 + Pr(2 sell) * 4 + Pr(3 sell) * 6 + Pr(4 sell) * 8 = 1/15 [1 * 2 + 2 * 4 + 3 * 6 + 9 * 8] = $ 6.67
Total expected cost = 4 * 1 = $ 4
Profit = $ 6.67 - $ 4 = $ 2.67
Let say he stocks 5 copies of the megazine
Total Expected earnings= Pr(1 sell) * 2 + Pr(2 sell) * 4 + Pr(3 sell) * 6 + Pr(4 sell) * 8 + Pr(5 sell) * 10 = 1/15 [1 * 2 + 2 * 4 + 3 * 6 + 4 * 8 + 5 * 10] = $ 7.33
Total expected cost = 5 * 1 = $ 5
Profit = $ 7.33 - $ 5 = $ 2.33
so we can say that profit is decreasing so we can say that the shopkeeper shall put 3 or 4 units in the stock.
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