Textbook Problem

A business executive, transferred from Chicago to Atlanta, needs to sell her house in Chicago quickly. The executive’s employer has offered to buy the house for $210,000, but the offer expires at the end of the week. The executive does not currently have a better offer but can afford to leave the house on the market for another month. From conversations with her realtor, the executive believes the price she will get by leaving the house on the market for another month is uniformly distributed between $200,000 and $225,000.

1. a. If she leaves the house on the market for another month, what is the mathematical expression for the probability density function of the sales price?
2. b. If she leaves it on the market for another month, what is the probability she will get at least $215,000 for the house?
3. c. If she leaves it on the market for another month, what is the probability she will get less than $210,000?
4. d. Should the executive leave the house on the market for another month? Why or why not?

To determine

Obtain the mathematical expression for the probability density function.

Answer to Problem 39SE

The probability density function for sales price is,

$f\left(x\right)=\{\begin{array}{l}\frac{1}{25,000}\text{for 200,000}\le x\le 250,000\\ 0\text{ elsewhere}\end{array}$

Explanation of Solution

Calculation:

The executive’s employer offers to buy a house for $210,000 and this offer will be expired by the end of the week. Post discussion with business executive’s relator, the seller believes that by leaving the house in the market for another month, the price for house will be uniformly distributed between $200,000 and $225,000.

The probability density function for uniform distribution is,

$f\left(x\right)=\{\begin{array}{l}\frac{1}{b-a}\text{for a}\le x\le b\\ 0\text{ elsewhere}\end{array}$

The probability density function for sales price is,

$f\left(x\right)=\{\begin{array}{l}\frac{1}{25,000}\text{for 200,000}\le x\le 250,000\\ 0\text{ elsewhere}\end{array}$

To determine

Find the probability that the house will get with at least $215,000.

Answer to Problem 39SE

The probability that the house will get with at least $215,000 is 0.4.

Explanation of Solution

Calculation:

The cumulative density function for uniform distribution is,

$P\left(x\le X\right)=\frac{X-a}{b-a}$

The probability that the house will get with at least $215,000 is,

$\begin{array}{c}P\left(x>215,000\right)=1-P\left(x\le 215,000\right)\\ =1-\frac{215,000-200,000}{25,000}\\ =1-\frac{15,000}{25,000}\\ =0.4\end{array}$

Thus, the value of $P\left(x>215,000\right)$ is 0.4.

To determine

Find the probability that the house will get less than $210,000.

Answer to Problem 39SE

The probability that the house will get less than $210,000 is 0.4.

Explanation of Solution

Calculation:

The probability that the house will get less than $210,000 is,

$\begin{array}{c}P\left(x<210,000\right)=\frac{210,000-200,000}{25,000}\\ =\frac{10,000}{25,000}\\ =0.4\end{array}$

Thus, the value of $P\left(x<210,000\right)$ is 0.4.

To determine

Explain whether the executive can leave the house in the market for another month or not.

Answer to Problem 39SE

The executive can leave the house in the market for another month with an excepted cost of $212,000.

Explanation of Solution

Calculation:

If the house is left in market for another month, the average sales price of house is,

$\begin{array}{c}E\left(x\right)=\frac{a+b}{2}\\ =\frac{200,000+225,000}{2}\\ =\frac{425,000}{2}\\ =212,500\end{array}$

This indicates that if the executive leaves the house in the market for another month, then the expected sales price is $2,500 higher than the before price $210,000. If the house is left in the market for another month, the executive will get less than the company’s offer with 0.4 probability. It represents the executive to leave the house for another month in the market with an expected cost of $215,000.