Question

Three coins are tossed; x = number of heads minus number of tails.

Short answer

The required values are mentioned below.

$\begin{array}{c}\mu =0\\ \mathrm{var}\left(x\right)=3\\ \sigma =\sqrt{3}\end{array}$.

Step-by-step solution

Given Information

Three coins are tossed.

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.

Let S be the sample space.

$S=\{HHH,HHT,HTH,HTT,THH,THT,THH,TT\}$

Find the random variable.

Find the value of .$x_1$

$\begin{array}{c}H=0\\ T=3\\ x_1=-3\\ p_1=1/8\end{array}$

Find the value of role="math" $x_2$.

$\begin{array}{c}H=1\\ T=2\\ x_2=-1\\ p_2=3/8\end{array}$

Find the value of $x_3$.

$\begin{array}{c}H=2\\ T=1\\ x_3=1\\ p_1=3/8\end{array}$

Find the value of $x_4$

$\begin{array}{c}H=3\\ T=0\\ x_4=3\\ p_4=1/8\end{array}$

The mean is given below.

$\begin{array}{c}\mu =\sum x_i{p}_i\\ \mu =-3\times \frac{1}{8}-\frac{3}{8}+\frac{3}{8}+3\times \frac{1}{8}\\ \mu =0\end{array}$

The variance is given below.

$\begin{array}{c}\mathrm{var}\left(x\right)=\sum {\left(x_i-\mu \right)}^2\\ p_i=\sum x_i^2{p}_i\\ =\times \frac{1}{8}\times \frac{3}{8}+\times \frac{3}{8}+\times \frac{1}{8}\\ \mathrm{var}\left(x\right)=3\end{array}$

The standard deviation is given below.

$\begin{array}{c}\sigma =\sqrt{\mathrm{var}\left(x\right)}\\ \sigma =\sqrt{3}\end{array}$

Cumulative function is given below.

$\begin{array}{c}{x}_1=-3\\ F(-3)=1/8\\ x_2=-1\\ F(-1)=1/2\end{array}$

Solve further.

$\begin{array}{c}{x}_3=1\\ F\left(1\right)=7/8\\ x_4=3\\ F\left(3\right)=1\end{array}$

Thegraph is shown below.