Probability Distributions and Expected Value
Understanding how random outcomes are distributed and what we can "expect" on average is one of the most powerful ideas in mathematics — with applications from insurance to game design to quantum physics.
1. Random Variables
Definition: A random variable $X$ is a function that assigns a numerical value to each outcome of a random experiment.
There are two types:
- Discrete Random Variable: Takes on a countable number of values (e.g., number of heads in 5 coin flips: $X \in \{0, 1, 2, 3, 4, 5\}$)
- Continuous Random Variable: Takes on any value in an interval (e.g., the exact height of a randomly selected student)
In this lesson, we focus primarily on discrete probability distributions.
2. Probability Distribution
Definition: A probability distribution is a table, formula, or graph that describes the probability of each possible value of a random variable.
For a discrete random variable $X$ with values $x_1, x_2, \ldots, x_n$, the probability distribution must satisfy:
Rule 1: $0 \leq P(X = x_i) \leq 1$ for all $i$
Rule 2: $\displaystyle\sum_{i=1}^{n} P(X = x_i) = 1$ (All probabilities sum to 1)
Example: Rolling a Fair Die
| $x$ | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| $P(X=x)$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $\frac{1}{6}$ |
Check: $\frac{1}{6} \times 6 = 1$ ✓