Chapter 15.3, Problem 12WE

a.

To determine

To calculate: The value of the equation $y=\frac{1}{\sqrt{2\pi }}{e}^{\left(-\frac{1}{2}\right)x^2}$ at $x=0$ .

Answer to Problem 12WE

The value of the equation $y=\frac{1}{\sqrt{2\pi }}{e}^{\left(-\frac{1}{2}\right)x^2}$ at $x=0$ is $0.3989$ .

Explanation of Solution

Given information:

The equation of standard normal curve is $y=\frac{1}{\sqrt{2\pi }}{e}^{\left(-\frac{1}{2}\right)x^2}$ .

Calculation:

Consider the equation of standard normal curve is $y=\frac{1}{\sqrt{2\pi }}{e}^{\left(-\frac{1}{2}\right)x^2}$ .

Substitute $x=0$ in the above equation.

  $\begin{array}{c}y=\frac{1}{\sqrt{2\pi }}{e}^{\left(-\frac{1}{2}\right){\left(0\right)}^2}\\ y=\frac{1}{\sqrt{2\pi }}{e}^0\\ y=\frac{1}{\sqrt{2\pi }}\end{array}$

Use a scientific calculator to compute the value,

  $\begin{array}{c}y=\frac{1}{\sqrt{2\pi }}\\ y=0.3989\end{array}$

Thus, the value of the equation $y=\frac{1}{\sqrt{2\pi }}{e}^{\left(-\frac{1}{2}\right)x^2}$ at $x=0$ is $0.3989$ .

b.

To determine

To calculate: The value of the equation $y=\frac{1}{\sqrt{2\pi }}{e}^{\left(-\frac{1}{2}\right)x^2}$ at $x=1$ .

Answer to Problem 12WE

The value of the equation $y=\frac{1}{\sqrt{2\pi }}{e}^{\left(-\frac{1}{2}\right)x^2}$ at $x=1$ is $0.24197$ .

Explanation of Solution

Given information:

The equation of standard normal curve is $y=\frac{1}{\sqrt{2\pi }}{e}^{\left(-\frac{1}{2}\right)x^2}$ .

Calculation:

Consider the equation of standard normal curve is $y=\frac{1}{\sqrt{2\pi }}{e}^{\left(-\frac{1}{2}\right)x^2}$ .

Substitute $x=1$ in the above equation.

  $\begin{array}{c}y=\frac{1}{\sqrt{2\pi }}{e}^{\left(-\frac{1}{2}\right){\left(1\right)}^2}\\ y=\frac{1}{\sqrt{2\pi }}{e}^{\left(-\frac{1}{2}\right)}\end{array}$

Use a scientific calculator to compute the value,

  $\begin{array}{c}y=\frac{1}{\sqrt{2\pi }}{e}^{\left(-\frac{1}{2}\right)}\\ y=0.24197\end{array}$

Thus, the value of the equation $y=\frac{1}{\sqrt{2\pi }}{e}^{\left(-\frac{1}{2}\right)x^2}$ at $x=1$ is $0.24197$ .

b.

To determine

To calculate: The value of the equation $y=\frac{1}{\sqrt{2\pi }}{e}^{\left(-\frac{1}{2}\right)x^2}$ at $x=2$ .

Answer to Problem 12WE

The value of the equation $y=\frac{1}{\sqrt{2\pi }}{e}^{\left(-\frac{1}{2}\right)x^2}$ at $x=2$ is $0.05399$ .

Explanation of Solution

Given information:

The equation of standard normal curve is $y=\frac{1}{\sqrt{2\pi }}{e}^{\left(-\frac{1}{2}\right)x^2}$ .

Calculation:

Consider the equation of standard normal curve is $y=\frac{1}{\sqrt{2\pi }}{e}^{\left(-\frac{1}{2}\right)x^2}$ .

Substitute $x=2$ in the above equation.

  $\begin{array}{c}y=\frac{1}{\sqrt{2\pi }}{e}^{\left(-\frac{1}{2}\right){\left(2\right)}^2}\\ y=\frac{1}{\sqrt{2\pi }}{e}^{\left(-\frac{4}{2}\right)}\\ y=\frac{1}{\sqrt{2\pi }}{e}^{\left(-2\right)}\end{array}$

Use a scientific calculator to compute the value,

  $\begin{array}{c}y=\frac{1}{\sqrt{2\pi }}{e}^{\left(-2\right)}\\ y=0.05399\end{array}$

Thus, the value of the equation $y=\frac{1}{\sqrt{2\pi }}{e}^{\left(-\frac{1}{2}\right)x^2}$ at $x=2$ is $0.05399$ .