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

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

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

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

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

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