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)	0.20	0.59	0.21

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

E(X)	= ft3
E(X2)	= ft6
V(X)	= ft6

(b) If the price of a freezer having capacity X cubic feet is 22X ? 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 22X ? 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 isX, 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

Answer 1

Concepts and reason

Mathematical expectation is the summation or mix of conceivable esteems from an irregular variable. It is also known as the product of the probability of an event occurring, denoted $P\left( {X = x} \right)$and the esteem relating with the genuine watched event of the occasion. The normal esteem is a valuable property of any irregular variable. Normally represented as $E\left( X \right)$the expect esteem can be registered by the summation general the particular esteems that the random variable can take.

Fundamentals

By the definition of mathematical expectation:

$E\left( X \right) = \sum\limits_x {x \times p\left( x \right)}$

$E\left( {{X^2}} \right) = \sum\limits_x {{x^2}p\left( x \right)}$

The variance of random variable is defined as follows:

${\rm{ V}}\left( X \right) = E\left( {{X^2}} \right) - {\left[ {E\left( X \right)} \right]^2}$

Properties of Expectation:

1.$E\left( {aX + b} \right) = aE\left( X \right) + b$

2.$E\left( {{\rm{constant}}} \right) = {\rm{constant}}$

3.$E\left( {aX} \right) = aE\left( X \right)$

4.$V\left( {aX + b} \right) = {a^2}V\left( X \right)$

5.$V\left( {{\rm{constant}}} \right) = 0$

(a)

The aim is to find $E\left( X \right),E\left( {{X^2}} \right)$and $V\left( X \right)$

(b)

If the price of a freezer having capacity X cubic feet is$\left( {22X{\rm{ + }}8.5} \right)$, compute the expected price paid by the next customer to buy a freezer: From part[a] $E\left( X \right) = 16.092$

Apply expectation to the function.

‎

(c)

If the price of a freezer having capacity X cubic feet is$\left( {22X{\rm{ + }}8.5} \right)$, compute the variance paid by the next customer to buy a freezer: From part[a] $V\left( X \right) = 3.2655$

Apply variance to the function.

$\begin{array}{c}\\V\left( {22X{\rm{ + }}8.5} \right) = V\left( {22X} \right) + V\left( {8.5} \right)\\\\ = {22^2}V\left( X \right) + 0\\\\ = {22^2} \times 3.2655\\\\ = 1580.502\\\end{array}$

(d)

If the price of a freezer having capacity function as $h\left( x \right) = X + 0.01{X^2}$, compute the expected value of the function. From part[a] $E\left( X \right) = 16.092$and $E\left( {{X^2}} \right) = 262.218$

Apply expectations to the function.

$\begin{array}{c}\\h\left( x \right) = X + 0.01{X^2}\\\\E\left[ {h\left( x \right)} \right] = E\left( {X + 0.01{X^2}} \right)\\\\ = E\left( X \right) + 0.01E\left( {{X^2}} \right)\\\\ = 16.092 + \left( {0.01 \times 262.218} \right)\\\\ = 16.092 + 2.6222\\\\ = 18.7142\\\end{array}$

Ans: Part a

The required values are$E\left( X \right) = 16.092$, $E\left( {{X^2}} \right) = 262.218$ and ${\rm{ V}}\left( X \right) = 3.2655$

Part b

The price of a freezer having the expected capacity as 362.524

Part c

The variance of the price $\left( {22X{\rm{ + }}8.5} \right)$ paid by the next customer is 1580.502

Part d

The expected actual capacity of the freezer purchased by the next customer for the function $h\left( x \right) = X + 0.01{X^2}$ is 18.7142