Chapter 7, Problem 117CE

a.

To determine

Compute the mean.

Answer to Problem 117CE

The mean is approximately $1,100K.

Explanation of Solution

Calculation:

It is given that the distribution of beach condominium prices (in $ thousands) follows triangular distribution with lower limit 500, mode 700 and upper limit of 2,100, that is, $T\left(500,700,2,100\right)$.

Triangular distribution:

A continuous random variable X is said to follow triangular distribution if the probability density function of X is,

$f\left(x\right)=\{\begin{array}{l}\frac{2\left(x-a\right)}{\left(b-a\right)\left(c-a\right)}\text{ for}a\le x\le b\\ \frac{2\left(c-x\right)}{\left(c-a\right)\left(c-b\right)}\text{ for}b\le x\le c\end{array}$ ,

where a, b and c are the lower limit, mode and upper limit, respectively.

Mean of a Triangular distribution is defined as,

$\mu =\frac{a+b+c}{3}$.

It is given that, $a=500$, $b=700$ and $c=2,100$.

Thus, the mean of the Triangular distribution is,

$\begin{array}{c}\mu =\frac{500+700+2,100}{3}\\ =\frac{3,300}{3}\\ =\underset{\_}{1,100}.\end{array}$

Therefore, the mean is $1,100K.

b.

To determine

Compute the standard deviation.

Answer to Problem 117CE

The standard deviation is $355.90K.

Explanation of Solution

Calculation:

The standard deviation of a Triangular distribution is defined as,

$\sigma =\sqrt{\frac{a^2+b^2+c^2-ab-ac-bc}{18}}$.

It is given that, $a=500$, $b=700$ and $c=2,100$.

Thus, the standard deviation of the Triangular distribution is,

$\begin{array}{c}\sigma =\sqrt{\frac{a^2+b^2+c^2-ab-ac-bc}{18}}\\ =\sqrt{\frac{{500}^2+{700}^2+2,{100}^2-\left(500\right)\left(700\right)-\left(500\right)\left(2,100\right)-\left(700\right)\left(2,100\right)}{18}}\\ =\sqrt{\frac{250,000+490,000+4,410,000-350,000-1,050,000-1,470,000}{18}}\\ =\sqrt{\frac{2,280,000}{18}}\\ =\sqrt{126,666.67}\\ \approx \underset{\_}{355.90}.\end{array}$

Therefore, the standard deviation of the Triangular distribution is $355.90K.

c.

To determine

Compute the probability that a condo price will be greater than $750K.

Answer to Problem 117CE

The probability that condo price will be greater than $750K is 0.8136.

Explanation of Solution

Calculation:

The cumulative distribution function (CDF) of a Triangular distribution is defined as,

$P\left(X\le x\right)=\{\begin{array}{l}\frac{{\left(x-a\right)}^2}{\left(b-a\right)\left(c-a\right)}\text{for }a\le x\le b\\ 1-\frac{{\left(c-x\right)}^2}{\left(c-a\right)\left(c-b\right)}\text{for b}\le x\le c\end{array}$ .

Assume that the random variable X denotes the beach condominium prices.

It is given that, $a=500$, $b=700$, $c=2,100$ and $x=750$.

Hence, the CDF of the Triangular distribution will be,

$P\left(X\le x\right)=1-\frac{{\left(c-x\right)}^2}{\left(c-a\right)\left(c-b\right)}\text{for b}\le x\le c$.

Thus, the probability that condo price will be greater than $750K is,

$\begin{array}{c}P\left(X>750\right)=\frac{{\left(2,100-750\right)}^2}{\left(2,100-500\right)\left(2,100-700\right)}\\ =\frac{1,{350}^2}{\left(1,600\right)\left(1,400\right)}\\ =\frac{1,822,500}{2,240,000}\\ =\underset{\_}{0.8136}.\end{array}$

Therefore, the probability that condo price will be greater than $750K is 0.8136.

d.

To determine

Draw the distribution and shade the area for the event in part (c).

Answer to Problem 117CE

The shaded sketch is obtained as:

Explanation of Solution

Calculation:

It is given that, $a=500$, $b=700$ and $c=2,100$.

Graphical Procedure:

Step by step procedure to obtain the sketch is given below:

• Draw a right skewed triangle with base from the lower limit 500 to upper limit 2,100 and with the height at the mode of 700.
• Shade the right side area from the mode of 750.

The shaded sketch is obtained as:

The shaded area in the sketch is the event in part (c).