Chapter 4.3, Problem 60E

(a)

To determine

To test: Whether the area under the curve is 1 or not.

Answer to Problem 60E

Solution: Yes, the area under the curve is 1.

Explanation of Solution

Calculation: The area of a triangle is,

$\text{Area}=\frac{1}{2}\times \text{base×height}$

The height is 1 and the base is equal to 2. Thus, the area can be calculated as:

$\begin{array}{c}\text{Area}=\frac{1}{2}\times \text{base×height}\\ \text{=}\frac{1}{\cancel{2}}\times \cancel{2}\times 1\\ =1\end{array}$

Hence, the area is 1. The provided curve is shown below:

(b)

Section 1:

To determine

To find: The probability $P\left(Y<1\right)$.

Answer to Problem 60E

Solution: The probability is 0.5.

Explanation of Solution

Calculation: First, obtain the area in the interval of $\left[0,1\right]$. Thus, from the provided triangle, base is 1 and height is equal to 1 because the triangle is symmetrical about its vertical line. Thus, the area can be calculated by the formula:

$\begin{array}{c}\text{Area}=\frac{1}{2}\times \text{base×height}\\ \end{array}$

In the graph, the total area is 1 and the dotted area which is also called the density of the curve. Thus, the area of $\Delta ACD$ or density of curve $Y<1$ can be calculated as:

$\begin{array}{c}\text{Density of }\left(Y<1\right)=\frac{1}{2}\times \text{base×height}\\ \text{=}\frac{1}{2}\times 1\times 1\\ =0.5\end{array}$

Now, the probability of Y is less than 1 can be calculated as:

$\begin{array}{c}P\left(Y<1\right)=\frac{\text{Density of }Y<1}{\text{Density of curve}}\\ =\frac{0.5}{1}\\ =0.5\end{array}$

Hence, the probability is 0.5.

Section 2:

To determine

To graph: The density curve and shade the area which represents the probability.

Explanation of Solution

Graph: In the provided triangle, draw a vertical line at 1. Construct a triangle on the left end of the triangle and shade the left area. Hence, the density curve is shown below:

Section 3:

To determine

To find: The area.

Answer to Problem 60E

Solution: The area is 0.5.

Explanation of Solution

Calculation: From the provided triangle, base is equal to 1 and height is equal to 1. Thus, the area can be calculated as:

$\begin{array}{c}\text{Area}=\frac{1}{2}\times \text{base×height}\\ \text{=}\frac{1}{2}\times 1\times 1\\ =0.5\end{array}$

Hence, the area is 0.5.

(c)

Section 1:

To determine

To find: The probability $P\left(Y>1.5\right)$.

Answer to Problem 60E

Solution: The probability is 0.875.

Explanation of Solution

First, obtain the area in the interval of $\left[0,1.5\right]$. Thus, from the triangle

base is equal to 0.5 and height is equal to 0.5. Thus, the area can be calculated by the formula:

$\text{Area}=\frac{1}{2}\times \text{base×height}$

The area of $\Delta BEF$ or the density of curve $Y>1.5$ can be calculated as:

$\begin{array}{c}\text{Area}=\frac{1}{2}\times \text{base×height}\\ \text{=}\frac{1}{2}\times 0.5\times 0.5\\ =0.125\end{array}$

Now, the probability of Y is greater than 1.5 can be calculated as:

$\begin{array}{c}P\left(Y>1.5\right)=1-P\left(Y\le 1.5\right)\\ =1-\frac{\text{Area of Δ}BEF}{\text{Density of curve}}\\ =1-\frac{0.125}{1}\\ =1-0.125\end{array}$

$=0.875$

Hence, the probability is 0.875.

Section 2:

To determine

To graph: The density curve and shade the area which represents the probability.

Explanation of Solution

Graph: In the provided triangle, draw a vertical line at 1.5. Construct a triangle on the right end of the triangle and shade the left area. Hence, the density curve is shown below:

Section 3:

To determine

To find: The area.

Answer to Problem 60E

Solution: The area is 0.125.

Explanation of Solution

Calculation: From the provided triangle, base is equal to 0.5 and height is equal to 0.5. Thus, the area can be calculated as:

$\begin{array}{c}\text{Area}=\frac{1}{2}\times \text{base×height}\\ \text{=}\frac{1}{2}\times 0.5\times 0.5\\ =0.125\end{array}$

Hence, the area is 0.125.

(d)

To determine

To find: The probability $P\left(Y>0.5\right)$.

Answer to Problem 60E

Solution: The probability is 0.875.

Explanation of Solution

Calculation: First, obtain the area in the interval of $\left[0,0.5\right]$. Thus, from the provided triangle, base is equal to 0.5 and height is equal to 0.5. Thus, the area can be calculated by the formula:

$\text{Area}=\frac{1}{2}\times \text{base×height}$

The area of $\Delta AGH$ can be calculated as:

$\begin{array}{c}\text{Area}=\frac{1}{2}\times \text{base×height}\\ \text{=}\frac{1}{2}\times 0.5\times 0.5\\ =0.125\end{array}$

Now, the probability of Y is greater than 0.5 can be calculated as:

$\begin{array}{c}P\left(Y>0.5\right)=1-P\left(Y\le 0.5\right)\\ =1-\frac{\text{Area of }\Delta AGH}{\text{Density of curve}}\\ =1-\frac{0.125}{1}\\ =1-0.125\end{array}$

$=0.875$

Hence, the probability is 0.875.