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

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

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

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.

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.

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

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

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.

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.

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

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

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.

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.

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.

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

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.

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

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

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.

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.

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.

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.

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.