---
title: 隨機變數 (Random Variable)
description: 
published: true
date: 2025-12-28T17:19:02.040Z
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
$P(X = x) = p_i$

Expectation
$E(X) = \sum_{i} p_i x_i$

### 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.

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

Expectation
$E(X) = \int xf(x) \,dx$

### Cumulative Distribution Function (CDF)

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

### Symmetric
If there is a point that,
$f(μ + x) = f(μ - x)$
Then,
$E(X) = μ$
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.
$F(X) = p$

### 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.

$Var(X) = E(X^{2}) - (E(X))^{2}$