Problem 7: [8 points] An appliance dealer sells three different models of uprigh

Question

Problem 7: [8 points] An appliance dealer sells three different models of upright freezers having 3, 5 and 9 cubic feet of storage space purchased by the next customer to buy a freezer. Suppose that X has p.mf. P(X = 3) = 0.2, P(X 5) 0.5 and P(X-9-c, where c is a constant in (0, 1]. (a) Find the value of c that makes this a valid p.m.f. Use that value from now [l (b) Derive the m.gf. M(t) of X. [2] (c) Using M(t) find E(x), E(x) and V(x). [1.,1,1] (d) If the price of a freezer having X cubic feet is 15xX2 +3X-2.5, what is the expected price paid by the next customer? 12

Explanation / Answer

(a)

Sum of probabilities should total to 1.

Hence, .2 + .5 + c = 1

c = .3

(b)

M(t) = .2xt , x=3

. 5xt. x=5

.3xt x = 9

It is a piece-wise constant function.

(c)

E(X) = M(1)  

= .2x , x=3   

. 5x. x=5

.3x x = 9

= .6 + 2.5 + 2.7

= 5.8

E(X2) = M (2)

= .2x2 , x=3   

. 5x2. x=5

.3x2 x = 9

= 1.8 + 12.5 + 24.3

= 38.6

V(x) = (E(x) )2 - E(x2)

= (5.8)2 - 38.6

= 33.64 - 38.6

= 4.96

(d) E(X) = Int ( 15X2 + 3X - 2.5 )

= 5 X3 + 1.5 X2 - 2.5X

This is the expected price paid by the next customer.

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