Random variables

An important concept in probability is a random variable. Oftentimes, we are not interested in the actual outcome of an experiment but some function of the outcome. For instance, let's define the function S to be the number of tails when flipping two coins. We know that Ω is {(H,H), (H,T), (T,H), (T,T)} where H stands for heads and T stands for tails. We also know that the probability of each of these outcomes is 1/4^th. However, I want to know the amount of tails in my outcomes, which I define as this:

If I define S to be a random variable, then:

In general, random variables are written with capital letters such as X, Y, and Z. Random variables are functions from the sample space Ω to a measurable space, which is not a trivial thing. Fortunately for us, for most of our random variables, the measurable space will be the real numbers.

Discrete random variables

There are continuous and discrete...