## Example

Find the expected value and variance of $X$ for the following probability distribution.

$x$ | 1 | 2 | 3 | 4 |
---|---|---|---|---|

$P(X=x)$ | 0.0625 | 0.1875 | 0.3125 | 0.4375 |

### Solution

The Expected value of $X$ is

```
$$
\begin{aligned}
E(X) &= \sum_x x*P(X=x)
\end{aligned}
$$
```

The expected value of $X^2$ is
```
$$
\begin{aligned}
E(X^2) &= \sum_x x^2*P(X=x)
\end{aligned}
$$
```

The variance of $X$ is
```
$$
\begin{aligned}
V(X) &= E[X-E(X)]^2 \\
&= E(X^2) - [E(X)]^2
\end{aligned}
$$
```

$x$ | $P(X=x)$ | $x*P(X=x)$ | $x^2*P(X=x)$ |
---|---|---|---|

1 | 0.0625 | 0.0625 | 0.0625 |

2 | 0.1875 | 0.375 | 0.75 |

3 | 0.3125 | 0.9375 | 2.8125 |

4 | 0.4375 | 1.75 | 7 |

```
$$
\begin{aligned}
\text{Mean } & =E(X)\\
& =\sum x*P(X=x) \\
& =(1*0.0625)+(2*0.1875)+(3*0.3125)+(4*0.4375) \\
& = 3.125
\end{aligned}
$$
```

```
$$
\begin{aligned}
E(X^2) & =\sum x^2*P(X=x) \\
& =(1*0.0625)+(4*0.1875)+(9*0.3125)+(16*0.4375)\\
& = 10.625
\end{aligned}
$$
```

```
$$
\begin{aligned}
V(X) &= E(X^2)-[E(X)]^2\\
&= 10.625 - [3.125]^2\\
&= 0.859375
\end{aligned}
$$
```

`$$sd = \sqrt{V(X)} =\sqrt{0.859375} = 0.9270248$$`