Question

Generalize problem $\mathbf{12}$to show that for the general binomial distribution $\mathbf{(}\mathbf{6}\mathbf{.}\mathbf{15}\mathbf{)}$ ,${\mathit{\sigma }}_{\mathbf{x}\mathbf{=}}\sqrt{\mathbf{n}\mathbf{p}\mathbf{q}}$and$\mathit{V}\mathit{a}\mathit{r}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{}\mathbf{=}\mathbf{}\mathit{n}\mathit{p}\mathit{q}$

Short answer

Result variance and standard deviation are mentioned below:

$$\begin{gathered} Var\left(x\right)=npq \\ \sigma =\sqrt{npq} \end{gathered}$$

Step-by-step solution

Given Information

$$\begin{gathered} p(x=0)=p \\ p(x=1)=q \\ p+q=1 \end{gathered}$$

Definition of Variance and Standard deviations

The standard deviation of a bunch of numbers is the distance between them and the mean. The variance is a measure of how much each point deviates from the mean on average.

Calculate expected value

Find expected value.

$\begin{array}{rcl}{\mu }_x& =& \underset{}{\sum x_{i}p\left(x_i\right)}\\ & =& 1\times p+0\times q\\ & =& p\\ & & \end{array}$

Calculate variance

Find variance.

$\begin{array}{rcl}Var\left(x\right)& =& \sum _1{\left(x-\mu \right)}^{2}p\left(x_i\right)\\ & =& \left(1-p^2\right)p+{\left(0-p\right)}^{2}q\\ & =& q^{2}p+p^{2}q\\ & =& qp\left(p+q\right)\end{array}$

$\begin{array}{rcl}Var\left(x\right)& =& pq\end{array}$

Calculate variance and standard deviation for nth experiment

Solve for variance.

$\begin{array}{rcl}Var\left(x\right)& =& Var(x_1+x_2+x_3+...+x_n)\\ & =& Var\left(x_1\right)+Var\left(x_2\right)+Var\left(x_3\right)+...+Var\left(x_n\right)\\ & =& Var\left(x\right)+Var\left(x\right)+Var\left(x\right)+...+Var\left(x\right)\\ & =& nVar\left(x\right)\\ Var\left(x\right)& =& npq\end{array}$

Solve for standard deviation.

$\begin{array}{rcl}\sigma & =& \sqrt{npq}\end{array}$

Hence, variance and standard deviation are mentioned below:

$\begin{array}{rcl}Var\left(x\right)& =& npq\\ \sigma & =& \sqrt{npq}\\ & & \end{array}$