Probability Distribution
Definition
A probability distribution assigns probabilities to events in a sample space. It is fully characterized by either:
- A Probability Mass Function (PMF) (discrete case), or
- A Probability Density Function (PDF) (continuous case).
Equivalently, by the cumulative distribution function:
$$ F(x) = \mathbb{P}(X \leq x). $$Properties of the CDF
- $F$ is non-decreasing and right-continuous.
- $\lim_{x \to -\infty} F(x) = 0$ and $\lim_{x \to \infty} F(x) = 1$.