Random Variables

A random variable $X$is basically an outcome of random event.

A random event is an experiment where the outcome of event can change.

All the different outcomes that are possible which the values random variable $X$ can take is called the support of the random variable.

A random variable is defined by its pdf.

PDF of a random variable $X$ defines what's the probability of each outcome being realized as the output of random event. So basically pdf is a function which assign all the possible outcome of random variable a probability value. And the the summation of all the probabilities values should be 1. i.e

Let $R_X$be the support of the random variable $X$i.e it's a set of all the values random variable $X$can take. And let $x \in R_X$

$$\sum_{x \in R_X}P(X=x) = 1$$

If the random variable is continuous then the summation above becomes integration.

Expected Value

Expected value or mean is same thing for a probability distribution and are used interchangeably.