# How do you find the mean of the random variable x?

## X= 5,10,15,20,25 P(x) = 1/5, 1/5 , 1/5, 1/5, 1/5 What is the variance and standard deviation of the random variable $x$? What is the standard deviation of the random variable $x$?

Jan 31, 2018

$\text{mean } E \left(X\right) = 15$

$V a r \left(X\right) = 50$

$s d = 7.07$

#### Explanation:

$\textcolor{red}{E \left(X\right) = {\sum}_{a l l x} x P \left(X = x\right)} - - - \left(1\right)$

in this case we have

$E \left(X\right) = 5 \times \frac{1}{5} + 10 \times \frac{1}{5} + 15 \times \frac{1}{5} + 20 \times \frac{1}{5} + 25 \times \frac{1}{5}$

$E \left(X\right) = 1 + 2 + 3 + 4 + 5$

$E \left(X\right) = 15$

the variance is calculated by

color(blue)(Var(X)=E(X^2)-E^"(X))---(2)

where $\textcolor{b l u e}{E \left({X}^{2}\right) = {\sum}_{a l l x} {x}^{2} P \left(X = x\right) - - \left(3\right)}$

$E \left({X}^{2}\right) = {5}^{2} \times \frac{1}{5} + {10}^{2} \times \frac{1}{5} + {15}^{2} \times \frac{1}{5} + {20}^{2} \times x \frac{1}{5} + {25}^{2} \times \frac{1}{5}$

E(X^")=1/5(25+100+225+400+625)

$E \left({X}^{2}\right) = 275.$

$V a r \left(X\right) = 275 - {15}^{2} = 275 - 225 = 50$

$s d = \sqrt{V} a r \left(X\right) = \sqrt{50} = 7.07$