Question
Find the expected value, variance and standard derivation of random variable $X$ whose probability mass function $(p.m.f.)$ is given below
$X =x$ $1$ $2$ $3$
$P ( X =x)$ $\frac{1}{5}$ $\frac{2}{5}$ $\frac{2}{5}$

Answer

$E(X)=\sum x_i P\left(x_i\right)$
$=1(1/5)+2(2/5)+3(2/5)$
$=(1+4+6)/5=11/5$
$=2.2$
$E\left(X^2\right)=\sum x_i^2 P\left(x_i\right)$
$=1^2(1/5)+2^2(2/5)+3^2(2/5)$
$=(1+8+18)/5=27/5$
$=5.4$
Var $(X) = E(X^2) - [E(X)]^2$
$= 5.4 - (2.2)^2$
$= 5.4 - 4.84$
$= 0.56$
$S.D. =\sqrt{\operatorname{Var}(X)}=\sqrt{0.56}=0.7483$ bnhyt $5r$

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