Question

If x is a binomial random variable, calculate , , and for each of the following:

1. n = 25, p = .5
2. n = 80, p = .2
3. n = 100, p = .6
4. n = 70, p = .9
5. n = 60, p = .8
6. n = 1000, p = .04

Short answer

1. $\mu$=12.5, ${\sigma }^2$=6.25, and$\sigma$ =2.5.
2. $\mu$=16, ${\sigma }^2$=12.8, and $\sigma$=3.58.
3. $\mu$=60,${\sigma }^2$=24, and $\sigma$=4.9.
4. $\mu$=63,${\sigma }^2$=6.3, and $\sigma$=7.94.
5. $\mu$=48,${\sigma }^2$=9.6, and $\sigma$=3.1.
6. $\mu$=40, ${\sigma }^2$=38.4, and$\sigma$ =6.2.

Step-by-step solution

Computation of  μ

a. The product of the value of n (number of trials), which is 25, and the value of p (number of successes), which is 0.5, will give the value of the mean$\mu$:

$\begin{array}{rcl}\mu & =& np\\ & =& 25\times 0.5\\ & =& 12.5\\ & & \end{array}$

The computed value of $\mathit{\mu }$is 12.5.

Computation of  σ2

The product of the value of n, which is 25, the value of p, which is 0.5, and the value of q, which is 0.5, will give the value of the variance ${\mathit{\sigma }}^{\mathbf{2}}$:

$\begin{array}{rcl}{\sigma }^2& =& npq\\ & =& np\left(1-p\right)\\ & =& \left(25\times 0.5\right)\left(1-0.5\right)\\ & =& 25\times 0.5\times 0.5\\ & =& 6.25\\ & & \end{array}$

The computed value of ${\sigma }^2$ is 6.25.

Computation of  σ

The square root of the value of${\sigma }^2$, which is 6.25, will give the value of the standard deviation$\sigma$:

role="math" localid="1655710063873" $\begin{array}{rcl}\sigma & =& \sqrt{{\sigma }^2}\\ & =& \sqrt{6.25}\\ & =& 2.5\\ & & \end{array}$

The computed value of$\mathit{\sigma }$is 2.5.

Computation of  μ

b. The product of the value of n, which is 80, and the value of p, which is 0.2, will give the value of $\mathit{\mu }$:

$\begin{array}{rcl}\mu & =& np\\ & =& 80\times 0.2\\ & =& 16\\ & & \end{array}$

The computed value of $\mathit{\mu }$is 16.

Computation of  σ2

The product of the value of n, which is 80, the value of p, which is 0.2, and the value of q, which is 0.8, will give the value of ${\sigma }^2$:

$\begin{array}{rcl}{\sigma }^2& =& npq\\ & =& np\left(1-p\right)\\ & =& \left(80\times 0.2\right)\left(1-0.2\right)\\ & =& 80\times 0.2\times 0.8\\ & =& 12.8\\ & & \end{array}$

Thus, the computed value of${\mathit{\sigma }}^{\mathbf{2}}$is 12.8.

Computation of  σ

The square root of the value of ${\sigma }^2$ , which is 6.25, will give the value of $\sigma$:

$\begin{array}{rcl}\sigma & =& \sqrt{{\sigma }^2}\\ & =& \sqrt{12.8}\\ & =& 3.58\\ & & \\ & & \end{array}$

The computed value of $\mathit{\sigma }$is 3.58.

Computation of  μ

c. The product of the value of n, which is 100, and the value of p, which is 0.6, will give the value of $\mu$:

$\begin{array}{rcl}\mathit{\mu }& \mathbf{=}& \mathbf{\text{np}}\\ & =& 100\times 0.6\\ & =& 60\\ & & \end{array}$

The computed value of$\mathit{\mu }$is 60.

Computation of  σ2

The product of the value of n, which is 100, the value of p, which is 0.6, and the value of q, which is 0.4, will give the value of ${\sigma }^2$:

$\begin{array}{rcl}{\sigma }^2& =& npq\\ & =& np\left(1-p\right)\\ & =& \left(100\times 0.6\right)\left(1-0.6\right)\\ & =& 100\times 0.6\times 0.4\\ & =& 24\\ & & \\ & & \end{array}$

Thus, the computed value of ${\sigma }^2$is 24.

Computation of  σ

The square root of the value of ${\mathit{\sigma }}^{\mathbf{2}}$, which is 24, will give the value of $\mathit{\sigma }$:

$\begin{array}{rcl}\sigma & =& \sqrt{{\sigma }^2}\\ & =& \sqrt{24}\\ & =& 4.9\\ & & \end{array}$

The computed value of$\sigma$is 4.9.

Computation of  

d. The product of the value of n, which is 70, and the value of p, which is 0.9, will give the value of $\mathit{\mu }$:

role="math" localid="1655707768889" $\begin{array}{rcl}\mathit{\mu }& \mathbf{=}& \mathit{n}\mathit{p}\\ & =& 70\times 0.9\\ & =& 63\\ & & \end{array}$

The computed value of$\mathit{\mu }$ is 63.

Computation of  

The product of the value of n, which is 70, the value of p, which is 0.9, and the value of q, which is 0.1, will give the value of ${\sigma }^2$:

$\begin{array}{rcl}{\sigma }^2& =& npq\\ & =& np\left(1-p\right)\\ & =& \left(70\times 0.9\right)\left(1-0.9\right)\\ & =& 70\times 0.9\times 0.1\\ & =& 6.3\\ & & \end{array}$

The computed value of ${\sigma }^2$is 6.3

Computation of  σ

The square root of the value of ${\mathit{\sigma }}^{\mathbf{2}}$, which is 6.3, will give the value of $\mathit{\sigma }$:

$\begin{array}{rcl}\sigma & =& \sqrt{{\sigma }^2}\\ & =& \sqrt{63}\\ & =& 7.94\\ & & \end{array}$

The computed value of$\mathit{\sigma }$ is 7.94.

Computation of  μ

e. The product of the value of n, which is 48, and the value of p, which is 0.8, will give the value of $\mu$:

$\begin{array}{rcl}\mu & =& np\\ & =& 60\times 0.8\\ & =& 48\\ & & \end{array}$

The computed value of$\mathit{\mu }$is 48.

Computation of  σ2

The product of the value of n, which is 48, the value of p, which is 0.8, and the value of q, which is 0.2, will give the value of ${\sigma }^2$:

$\begin{array}{rcl}{\sigma }^2& =& \mathit{n}\mathit{p}\mathit{q}\\ & \mathbf{=}& \mathit{n}\mathit{p}\left(1-p\right)\\ & =& \left(60\times 0.8\right)\left(1-0.8\right)\\ & =& 60\times 0.8\times 0.2\\ & =& 9.6\\ & & \end{array}$

The computed value of${\sigma }^2$is 9.6.

Computation of  σ

The square root of the value of ${\sigma }^2$, which is 9.6, will give the value of $\sigma$:

$\begin{array}{rcl}\sigma & =& \sqrt{{\sigma }^2}\\ & =& \sqrt{9.6}\\ & =& 3.1\\ & & \end{array}$

The computed value of$\mathit{\sigma }$ is 3.1.

Computation of  μ

The product of the value of n, which is 1000, and the value of p, which is 0.04, will give the value of $\mathit{\mu }$:

$\begin{array}{rcl}\mathit{\mu }& \mathbf{=}& \mathit{n}\mathit{p}\\ & =& 1000\times 0.04\\ & =& 40\\ & & \end{array}$

The computed value of $\mathit{\mu }$ is 40.

Computation of  σ2

The product of the value of n, which is 1000, the value of p, which is 0.04, and the value of q, which is 0.96, will give the value of ${\sigma }^2$:

$\begin{array}{rcl}{\sigma }^2& =& \mathit{n}\mathit{p}\mathit{q}\\ & \mathbf{=}& \mathit{n}\mathit{p}\left(1-p\right)\\ & =& 1000\times 0.04\left(1-0.04\right)\\ & =& 1000\times 0.04\times 0.96\\ & =& 38.4\\ & & \end{array}$

The computed value of ${\sigma }^2$is 38.4.

Computation of  σ 

The square root of the value of ${\sigma }^2$, which is 38.4, will give the value of $\sigma$:

role="math" localid="1655702205065" $\begin{array}{rcl}\sigma & =& \sqrt{{\sigma }^2}\\ & =& \sqrt{38.4}\\ & =& 6.2\\ & & \end{array}$

The computed value of $\sigma$is 6.2.