Random Variables

When the numerical value of a variable is determined by a chance event, that variable is called a random variable.
Discrete and Continuous Random Variables

Definition: A random variable is a function that assigns a real number $X(s)$ to every element $s\in S$. Where $S$ is the sample space corresponding to a random experiment $E$.

Random variables can be discrete or continuous.

Discrete Random Variable:

Discrete random variables take on integer values, usually the result of counting.
Example: We flip a coin and count the number of heads. The number of heads results from a random process - flipping a coin. And the number of heads is represented by an integer value - a number between 0 and plus infinity. Therefore, the number of heads is a discrete random variable.


Continuous Random Variable:

Continuous random variables, in contrast, can take on any value within a range of values.
Example: We flip a coin many times and compute the average number of heads per flip. The average number of heads per flip results from a random process - flipping a coin. And the average number of heads per flip can take on any value between 0 and 1, even a non-integer value. Therefore, the average number of heads per flip is a continuous random variable.