---
title: 隨機變數 (Random Variable)
description: 
published: true
date: 2025-12-28T19:20:15.044Z
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 ≤ p~i~ ≤ 1 and ∑ p~i~ = 1**

Probability  
 $$\\leq$$

Expectation  
 

### Cumulative Distribution Function (CDF)

  
 

## Continuous

### Probability Density Function (PDF)

Probabilistic properties of a continuous random variable.

Expectation  
 

### Cumulative Distribution Function (CDF)

  
  
  
 

### Symmetric

If there is a point that,  
  
Then,  
  
Is the expectation of this random variable, equal to the point of symmetry.

### Median and Quantiles

The middle value of the random variable.  
For median, set p to 0.5.  
 

### Variance

A positive quantity that measures the spread of the distribution of the random variable about its mean value.  
Larger values of the variance indicate that the distribution is more spread out.