Question

How to find the mean of a probability distribution?

Answer 1

That depends. If you have a finite data set, you would add up all the points you have and divide by the total count.

Or, if you are working with pure distributions, the mean is the same as the expected value of the corresponding random variable.

Suppose you have a discrete random variable $X$ with a given probability mass function $f_X(x)$, then the mean is given by

$\mathbb E(X)=\displaystyle\sum_xxf_X(x)$

which would mean you take all the possible probability for the event that $X=x$, multiply each by that $x$, and add them together.

If the distribution is continuous, say a random variable $Y$ that has probability distribution function $f_Y(y)$ over some support $S$, then the mean is

$\mathbb E(Y)=\displaystyle\int_Syf_Y(y)\,\mathrm dy$