8.2 Theory of Random Variable

A Random Variable (\(X\)) is a real-valued function whose outcome we don’t know beforehand.

I am going to flip a fair coin 3 times (the coin has 2-sides, we’ll call one side Heads and the other Tails).

Assume there are only 2 possible outcomes and P(Heads) = P(Tails) = 0.5 (can’t land on its side).
Below are three possible random variables based on the same random process (flipping a 2-sided coin 3 times):
Example 1 - \(X\): the number of heads in 3 coin flips
What are the possible values of \(X\)?
Example 2 - Say I give you 3 dollars for each head
\(Y\): the amount of money won from 3 coin flips, \(Y = 3*X\)
Example 3 - \(Z\): the number of heads on the last flip of 3 coin flips
The possible values are 0 or 1.

A probability model for random variable \(X\) gives the possible values of \(X\) and the associated probabilities.