Question

An certain brand of upright freezer is available in three different rated capacities: 16 ft3, 18 ft3, 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.

x 16 18 20
p(x) 0.3 0.1 0.6

(a) Compute E(X), E(X2), and V(X).
(b) If the price of a freezer having capacity X is 60X − 650, what is the expected price paid by the next customer to buy a freezer?

Answer 1

`E(X)=16\cdot 0.3+18\cdot 0.1+20\cdot 0.6=18.6`

`E(X^2)=16^2\cdot 0.3+18^2\cdot 0.1+20^2\cdot 0.6=349.2`

`V(X) = E(X^2)-[E(X)]^2=349.2-(18.6)^2=3.24`

The expected price paid by the next customer to buy a freezer is $466

Explanation:

From the information given we know the probability mass function (pmf) of random variable X.

`\left|\begin{array}cx&16&18&20\\p(x)&0.3&0.1&0.6\end{array}\right|`

Point a:

• The Expected value or the mean value of X with set of possible values D, denoted by E(X) or μ is

`E(X) = $\sum_(x\in D) x\cdot p(x)`

Therefore

`E(X)=16\cdot 0.3+18\cdot 0.1+20\cdot 0.6=18.6`

• If the random variable X has a set of possible values D and a probability mass function, then the expected value of any function h(X), denoted by E[h(X)] is computed by

`E[h(X)] = $\sum_(D) h(x)\cdot p(x)`

So
`h(X) = X^2` and

`E[h(X)] = $\sum_(D) h(x)\cdot p(x)\\E[X^2]=$\sum_(D)x^2\cdot p(x)\\ E(X^2)=16^2\cdot 0.3+18^2\cdot 0.1+20^2\cdot 0.6\\E(X^2)=349.2`

• The variance of X, denoted by V(X), is

`V(X) = $\sum_(D)E[(X-\mu)^2]=E(X^2)-[E(X)]^2`

Therefore

`V(X) = E(X^2)-[E(X)]^2\\V(X)=349.2-(18.6)^2\\V(X)=3.24`

Point b:

We know that the price of a freezer having capacity X is 60X − 650, to find the expected price paid by the next customer to buy a freezer you need to:

From the rules of expected value this proposition is true:

`E(aX+b)=a\cdot E(x)+b`

We have a = 60, b = -650, and E(X) = 18.6. Therefore

The expected price paid by the next customer is

`60\cdot E(X)-650=60\cdot 18.6-650=466`