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. x 13.5 15.9 19.1 p(x) 0.17 0.57 0.26 (a) Compute E(X), E(X2), and V(X). (Round your answers to two decimal places.) E(X) = 16.33 Correct: Your answer is correct. ft3 E(X2) = 269.94 Correct: Your answer is correct. ft6 V(X) = 3.46 Correct: Your answer is correct. ft6 (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.) 448.74 Correct: Your answer is correct. 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 − 0.02X2. What is the expected actual capacity of the freezer purchased by the next customer? (Round your answer to three decimal places.) 10.93 Correct: Your answer is correct. ft3

Answer 1

Answer:

Part a:The value of E(X) is 16.33, E(X^2) is 269.94 and value of V(X) is 3.46.

Part b:The expected price of a freezer of capacity X is $448.74.

Part c:The variance of the price of a freezer of capacity X is $2496.

Part d:The expected actual capacity of the freezer is 10.93

Step-by-step explanation:

From the data, the values of X and p(x) are as

  X  = 13.5 | 15.9 | 19.1

p(x) = 0.17 | 0.57 | 0.26

Part a

The values are as follows:

E(X)

$E(x)=\sum x p(x)\\E(x)= [(13.5)(0.17)+(15.9)(0.57)+(19.1)(0.26)]\\E(x)= 16.33$

E(X^2)

$E(x^2)=\sum x^2 p(x)\\E(x^2)= [(13.5)^2(0.17)+(15.9)^2(0.57)+(19.1)^2(0.26)]\\E(x^2)= 269.94$

V(X)

$V(X)=E(X^2)-(E(X))^2\\V(X)=3.46$

So the value of E(X) is 16.33, E(X^2) is 269.94 and value of V(X) is 3.46.

Part b:

Expected price is given as

$E(28X-8.5)=E(28X)-8.5\\E(28X-8.5)=28E(X)-8.5\\E(28X-8.5)=28(16.33)-8.5\\E(28X-8.5)=\ 448.74$

So the expected price of a freezer of capacity X is $448.74.

Part c:

Variance is given as

$V(28X-8.5)=V(28X)\\V(28X-8.5)=28^2V(X)\\V(28X-8.5)=28^2(3.46)\\V(28X-8.5)=\ 2496$

So the variance of the price of a freezer of capacity X is $2496.

Part d:

The expected actual capacity is given as

$h(X)=X-0.02X^2\\E(h(X))=E(X-0.02X^2)\\E(h(X))=E(X)-E(0.02X^2)\\E(h(X))=E(X)-0.02E(X^2)\\E(h(X))=16.33-0.02(269.94)\\E(h(X))=10.93$

So the expected actual capacity of the freezer is 10.93