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

## Discrete

### Probability Mass Function (PMF)

A set of probability value **p~i~** assigned to each of the values taken by the discrete random variables **x~i~**.

$0 \leq p_i \leq 1$ 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)

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