how to find the variance and standard deviation of a discrete random variable

Let X be a discrete random variable, such that

$P(X=x_1)=p_1$

$P(X=x_2)=p_2$

...

$P(X=x_n)=p_n$

Then its variance can be found as $V(X)=E(X^2)-E^2(X)$ , where E(X), E($X^2$) - first and second central moment respectively, so

$E(X)=\displaystyle\sum_{i=1}^np_i*x_i$

$E(X^2 )=\displaystyle\sum_{i=1}^np_i*x^2_i$

Discrete random variable can take infinite(countable) amount of values, in that case sum will be from 1 to infinity, and the point is to find the sum of the infinite row

Standard deviation can be found as a square root of the variance, so

$\sigma(X)=\sqrt{V(X)}$