Question

A certain brand of upright freezer is available in three different rated capacities: 16 ft3, 18 ft3, and 20 ft3. Let X 5 the rated capacity of a freezer of this brand sold at a certain store. Suppose that X has pmf x 16 18 20 p(x) .2 .5 .3 a. Compute E(X), E(X2 ), and V(X). b. If the price of a freezer having capacity X is 70X 2 650, 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?

Answer 1

To calculate expected values and variance, apply probability mass function values to the defined formulas. The expected value of the rated capacity X is 18.2 ft3, the variance V(X) is 1.96 ft3^2, and the expected price paid by the next customer for a freezer is $624. The variance of the price V(Price) is $9604.

Step-by-step explanation:

Expected Value, Variance, and Price Calculation

Let the random variable X represent the rated capacity of a freezer. The probability mass function (pmf) of X is given as p(16)=0.2, p(18)=0.5, and p(20)=0.3. To compute the expected value E(X), we use the formula E(X)=Σx*p(x), summing over all possible values of X:

E(X) = 16*0.2 + 18*0.5 + 20*0.3 = 3.2 + 9 + 6 = 18.2 ft3

To compute the second moment E(X2), we square the values of X and multiply by their probabilities:

E(X2) = 162*0.2 + 182*0.5 + 202*0.3 = 51.2 + 162 + 120 = 333.2

The variance V(X) is given by V(X) = E(X2) - [E(X)]2:

V(X) = 333.2 - (18.2)2 = 333.2 - 331.24 = 1.96 ft32

For the expected price: Price = 70X - 650, we can find the expected price E(Price) by substituting E(X) into the price formula:

E(Price) = 70*E(X) - 650 = 70*18.2 - 650 = 1274 - 650 = $624

The variance of the price is not simply V(Price) = V(70X - 650) because the subtraction of 650 does not affect the variance. Therefore, V(Price) is equal to 702 times the variance of X:

V(Price) = (70)2*V(X) = 4900*1.96 = 9604