Numerical Characteristics of Probability Distributions
Big Idea: Instead of describing an entire probability distribution, we can summarize it using a few key numbers — its numerical characteristics. These numbers tell us about the "center," "spread," and "shape" of the distribution.
1. Mathematical Expectation (Expected Value)
Definition
The mathematical expectation (or expected value) of a discrete random variable $X$ is the weighted average of all possible values, where each value is weighted by its probability:
$$ E(X) = \sum_{i=1}^{n} x_i \cdot P(X = x_i) = x_1 p_1 + x_2 p_2 + \cdots + x_n p_n $$
where $x_1, x_2, \ldots, x_n$ are the possible values and $p_1, p_2, \ldots, p_n$ are their corresponding probabilities.
Intuition: Think of $E(X)$ as the "balance point" of the distribution — if you placed weights $p_i$ at positions $x_i$ on a number line, $E(X)$ is where the line would balance perfectly.
Properties of Expected Value
- $E(c) = c$ where $c$ is a constant
- $E(cX) = c \cdot E(X)$ — constants factor out
- $E(X + Y) = E(X) + E(Y)$ — expectation is linear
- $E(X + c) = E(X) + c$ — shifting values shifts the mean