Question

A card is drawn from a shuffled deck. Let $x=10$if it is an ace or a face card; $\mathit{x}\mathbf{=}\mathbf{-}\mathbf{1}$if it is a$2$ ; and$x=0$ otherwise.

Short answer

The required values are mentioned below.

$\begin{array}{c}\mu =3\\ \mathrm{var}\left(x\right)=\frac{284}{13}\\ \sigma =4.67\end{array}$

Step-by-step solution

Given Information

A card is drawn from a shuffled deck.

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.

$\begin{array}{c}\mu =\sum x_i{P}_i\\ \mu =\frac{10\times 16}{52}-\frac{1\times 4}{52}\\ \mu =3\end{array}$

The variance is given below.

$\begin{array}{c}\mathrm{var}\left(x\right)=\sum {\left(x_i-\mu \right)}^{2}p\left(x_i\right)\\ \mathrm{var}\left(x\right)={\left(10-3\right)}^2\times \frac{16}{52}+{\left(-1-3\right)}^2\times \frac{4}{52}+3^2\times \frac{32}{52}\\ \mathrm{var}\left(x\right)=\frac{284}{13}\end{array}$

The standard deviation is given below.

$\begin{array}{c}\sigma =\sqrt{\mathrm{var}\left(x\right)}\\ \sigma =\sqrt{\frac{284}{13}}\\ \sigma =4.67\end{array}$

Thegraph is shown below.

Hence, the required values are mentioned below.

$\begin{array}{c}\mu =3\\ \mathrm{var}\left(x\right)=\frac{284}{13}\\ \sigma =4.67\end{array}$