An certain brand of upright freezer is available in three different rated capaci

Question

An certain brand of upright freezer is available in three different rated capacities: 16 ft3, 18 ft, and 20 ft3. Let X = the rated capacity of a freezer of this brand sold at a certain store. Suppose that X has the following pmf. 161820 p(x) 0.3 0.1 0.6 (a) Compute E(X), E(X2), and V(x) E(X) = | E(X2) = I ft (b) If the price of a freezer having capacity X is 61X-650, what is the expected price paid by the next customer to buy a freezer? (c) What is the variance of the price paid by the next customer? (d) Suppose that although the rated capacity of a freezer is X, the actual capacity is h(X) - x - 0.009x2. What is the expected actual capacity of the freezer purchased by the next customer? ft3

Explanation / Answer

a) E(X) = 16 * 0.3 + 18 * 0.1 + 20 * 0.6 = 18.6

E(X2) = 162 * 0.3 + 182 * 0.1 + 202 * 0.6 = 349.2

V(X) = E(X2) - (E(X))2 = 349.2 - 18.62 = 3.24

b) Expected value = E(61X - 650) = 61 * E(X) - 650 = 61 * 18.6 - 650 = 484.6

c) Variance = V(61X - 650) = 612 * V(X) = 612 * 3.24 = 12056.04

d) E(h(X)) = E(X - 0.009X2) = E(X) - 0.009 * E(X2) = 18.6 - 0.009 * 349.2 = 15.4572

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