What is a random variable? Section
Let's use a scenario to introduce the idea of a random variable.
Suppose we flip a fair coin three times and record if it shows a head or a tail. The outcome or sample space is S={HHH,HHT,HTH,THH,TTT,TTH,THT,HTT}. There are eight possible outcomes and each of the outcomes is equally likely. Now, suppose we flipped a fair coin four times. How many possible outcomes are there? There are $2^4 = 16$. How about ten times? $1024$ possible outcomes! Instead of considering all the possible outcomes, we can consider assigning the variable $X$, say, to be the number of heads in $n$ flips of a fair coin. If we flipped the coin $n=3$ times (as above), then $X$ can take on possible values of \(0, 1, 2,\) or \(3\). By defining the variable, \(X\), as we have, we created a random variable.
 Random Variable

A random variable is a variable that takes on different values determined by chance. In other words, it is a numerical quantity that varies at random.
Types of Random Variables
There are mainly two types of random variables:
 Discrete Random Variable
 When the random variable can assume only a countable, sometimes infinite, number of values.
 Continuous Random Variable
 When the random variable can assume an uncountable number of values in a line interval.
Probability Functions
Transforming the outcomes to a random variable allows us to quantify the outcomes and determine certain characteristics. If we have a random variable, we can find it’s probability function.
 Probability Function

A probability function is a mathematical function that provides probabilities for the possible outcomes of the random variable, \(X\). It is typically denoted as \(f(x)\).
There are two classes of probability functions: Probability Mass Functions and Probability Density Functions.
 Probability Mass Function (PMF)

If the random variable is a discrete random variable, the probability function is usually called the probability mass function (PMF). If X is discrete, then \(f(x)=P(X=x)\). In other words, the PMF for a constant, \(x\), is the probability that the random variable \(X\) is equal to \(x\). The PMF can be in the form of an equation or it can be in the form of a table.

Properties of probability mass functions:
 \(f(x)>0\), for x in the sample space and 0 otherwise.
 \(\sum_x f(x)=1\). In other words, the sum of all the probabilities of all the possible outcomes of an experiment is equal to 1.

 Probability Density Function (PDF)

If the random variable is a continuous random variable, the probability function is usually called the probability density function (PDF). Contrary to the discrete case, $f(x)\ne P(X=x)$

Properties of a probability density function:
 \(f(x)>0\), for x in the sample space and 0 otherwise.
 The area under the curve is equal to 1.

The probability of a random variable being less than or equal to a given value is calculated using another probability function called the cumulative distribution function.
 Cumulative Distribution Function (CDF)

A cumulative distribution function (CDF), usually denoted $F(x)$, is a function that gives the probability that the random variable, X, is less than or equal to the value x.

\(F(x)=P(X\le x)\)

Note! The definition of the cumulative distribution function is the same for a discrete random variable or a continuous random variable. For a continuous random variable, however, \(P(X=x)=0\). Therefore, the CDF, \(F(x)=P(X\le x)=P(X<x)\), for the continuous case.
