An appliance dealer sells three different models of upright freezers having 13.5

Question

An appliance dealer sells three different models of upright freezers having 13.5, 15.9, and 19.1 cubic feet of storage space, respectively. Let X = the amount of storage space purchased by the next customer to buy a freezer. Suppose that X has the following pmf.

(a) Compute E(X), E(X2), and V(X). (Round your answers to two decimal places.)

(b) If the price of a freezer having capacity X cubic feet is 28X 8.5, what is the expected price paid by the next customer to buy a freezer? (Round your answer to the nearest whole number.)
dollars

(c) What is the variance of the price 28X 8.5 paid by the next customer? (Round your answer to the nearest whole number.)
dollars2

(d) Suppose that although the rated capacity of a freezer is X, the actual capacity is h(X) = X 0.01X2. What is the expected actual capacity of the freezer purchased by the next customer? (Round your answer to three decimal places.)
ft3

Explanation / Answer

Note : Preparing table will give best understanding for this soultion.

Solution:

Table for easy computation for vlaues of E(X), E(X2), and V(X)

Answer (a)

Compute E(X), E(X2), and V(X) is given as:

Formula to calculate E(X) = (x).p(x)                                              [i.e. sum of all the product values of (x).p(x)]

E(X) = (13.5)(0.17) + (15.9)(0.52) + (19.1)(0.31)

E(X) = 2.295+8.268+5.921 = 16.48   (rounded to two decimal places)

Formula to calculate: E(X)2 = (x2).p(x)

E(X)2 = (13.5)2(0.17) + (15.9)2(0.52) + (19.1)2(0.31) = 30.9825+131.4612+113.0911 = 275.53 (rounded to two decimal places)

Formula to calculate: V(X) = E(X2)- E(X)2

putting values of E(X2)and E(X)2 in the above forumula to get V(X)

V(X) = 275.53 - 271.72 = 3.81                                                                  [E(X)2= (16.484)2 = 271.72]

Answer (b)

If the price of a freezer having capacity X cubic feet is 28X 8.5, then the expected price paid by the next customer to buy a freezer would be given by

E(28X-8.5) = 28E(X) - 8.5 = (28) (16.484) - (8.5) = 461.692 - 8.5 = 453.192

= 453 dollars (answer rounded to the nearest whole number).

Answer (c) The variance of the price 28X 8.5 paid by the next customer would be given by V(28X-8.5)

V(28X) = (28)2 V(X) = (784)(3.812) = 2989                                                              [put V(X) = 3.812]

dollars2

Answer (d)

The expected actual capacity of the freezer purchased by the next customer would be given by

E(h(x) = E[X-.01X2] = E(X) - .01E(X2) = 16.484 - (.01) (275.5348) = 16.484 - 2.755 = 13.7286 = 13.729 (rounded to three decimal places).

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