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# 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.