wiki/education/statistics/random-variable.md

title, description, published, date, tags, editor, dateCreated

title	description	published	date	tags	editor	dateCreated
隨機變數 (Random Variable)		true	2025-12-28T17:12:39.067Z		markdown	2025-12-28T16:41:51.938Z

Discrete

Probability Mass Function (PMF)

A set of probability value pi assigned to each of the values taken by the discrete random variables xi.

0 \leq p_i \leq 1 and \sum_{p_i} = 1

Probability P(X = x) = p_i

Cumulative Distribution Function (CDF)

F(X) = P(X \leq x) F(X) = \sum_{y : y \leq x} P(X = y)

Continuous

Probability Density Function (PDF)

Probabilistic properties of a continuous random variable.

\int f(x) \,dx\ = 1

Cumulative Distribution Function (CDF)

F(X) = P(X \leq x) = \int_{-\infty}^{x} f(y) \,dy\\ f(X) = \frac{dF(x)}{dx} P(a < x \leq b) = P(X \leq b) - P(X \leq a) = F(b) - F(a) P(a < x \leq b) = P(a \leq x \leq b)