Question

A random variable x takes the values with probabilities $\frac{5}{12},\frac{1}{3},\frac{1}{12},\frac{1}{6}$.

Short answer

The required values are mentioned below.

$\begin{array}{c}\mu =1\\ \mathrm{var}\left(x\right)=\frac{7}{6}\\ \sigma =\sqrt{\frac{7}{6}}\end{array}$

Step-by-step solution

Given Information

A random variable x.

Definition of the cumulative distribution function

The likelihood that a comparable continuous random variable has a value less than or equal to the function's argument is the value of the function.

Find the values

The mean is given below.

$\begin{array}{c}\mu =\sum _{0\times 5}p\left(x_i\right)\\ \mu =\frac{0\times 5}{12}+\frac{1\times 1}{3}+\frac{2\times 1}{12}+\frac{3\times 1}{6}\\ \mu =1\end{array}$

The variance is given below.

$\begin{array}{c}\mathrm{var}\left(x\right)=\sum {\left(x_i-\mu \right)}^{2}p\left(x_i\right)\\ \mathrm{var}\left(x\right)={\left(0-1\right)}^2\times \frac{5}{12}+{\left(1-1\right)}^2\times \frac{1}{3}+\times \frac{1}{12}+{\left(3-1\right)}^2\times \frac{1}{6}\\ \mathrm{var}\left(x\right)=\frac{7}{6}\end{array}$

The standard deviation is given below.

$\begin{array}{c}\sigma =\sqrt{\mathrm{var}\left(x\right)}\\ \sigma =\sqrt{\frac{7}{6}}\end{array}$

Thegraph is shown below.

Hence, the required values are mentioned below.

$\begin{array}{c}\mu =1\\ \mathrm{var}\left(x\right)=\frac{7}{6}\\ \sigma =\sqrt{\frac{7}{6}}\end{array}$