# Probability_Theory

This blog just works as a formula stack.

## Formula

### Law of Total Probability

IF: {${A_{i}:i=1,2,3…,n}$} is a finite or countably infinite partition of a sample space.

THEN for any event $B$:
$$P(B)=\sum_{i=1}^n P(A_{i})P(B|A_{i})$$

### Bayes’ Theorem

IF:{${A_{i}:i=1,2,3,…,n}$} is a finite or countably infinite partition of a sample space (happens firstly), and $B$ is a fixed event(happens secondly).

THEN for any event $A_{k}(k\in{1,2,3,…,n})$:
$$P(A_{k}|B)=\frac{P(A_{k})P(B|A_{k})}{\sum_{i=1}^nP(A_{i})P(B|A_{i})}$$

### Binomial Distribution $X \sim B(n, p)$

IF the random variable $X$ follows the binomial distribution with and $p ∈ [0,1]$, we write $X \sim B(n, p)$.

THENThe probability of getting exactly k successes in n independent Bernoulli trials is given by the probability mass function :
$$P\left(X=k\right)=C_{n}^kp^k(1-p)^k$$

### Poisson Distribution $X \sim P(\lambda)$

IF a discrete random variable X is said to have a Poisson distribution, with parameter $\lambda>0$, we write $X \sim P(\lambda)$ or $X \sim \pi(\lambda)$.

THEN it has a probability mass function given by :
$$P\left(X=k\right)=\frac{\lambda^k}{k!}e^{-\lambda}$$

### Continuous Uniform Distribution $X \sim U(a,b)$

IF the probability density function of the continuous uniform distribution $x$ is :
$$f(x) = \begin{cases} \frac{1}{b-a},& a<x<b\\ 0,& else \\ \end{cases}$$
THEN we write $X \sim U(a,b)$, and the cumulative distribution function is :
$$F(x) = \begin{cases} 0,& x<a\\ \frac{x-a}{b-a},& a \leq x < b \\ 1,& x \geq b \end{cases}$$

### Exponential distribution $X \sim E(\lambda)$

IF the probability density function of the continuous uniform distribution $x$ and the rate parameter $\lambda > 0$ is :
$$f(x;\lambda) = \begin{cases} \lambda e^{-\lambda x},& x\geq 0 \\ 0,& x<0 \\ \end{cases}$$
THEN we write $X \sim E(\lambda)$, and the cumulative distribution function is given by :
$$F(x)=\begin{cases} 1-e^{-\lambda x},&x \geq 0\\ 0,&x<0\\ \end{cases}$$

### Normal Distribution $X \sim N(\mu,\sigma^2)$

Normal distribution, also called Gaussian distribution.

IF there is a real-valued random variable X, and the general form of its probability density function is:
$$f(x)=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$
THEN We write $X \sim N(\mu,\sigma^2)$. The parameter $\mu$ is the mean or expectation of the distribution (and also its median and mode), while the parameter $\sigma$ is its standard deviation. The variance of the distribution is $\sigma^2$.

### Standard Normal Distribution $X \sim N(0,1)$

IF $X \sim N(\mu,\sigma^2)$.

THEN when $\mu=0,\sigma=1$, we write $X \sim N(0,1)$. It is described by this probability density function:
$$\varphi(x) = \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}$$

and the cumulative distribution function is given by :
$$\phi(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{-\frac{t^2}{2}} dt$$

To Be Continued…

Probability_Theory
http://xxblog.net/Mathematics/Probability-Theory/
Author
XX
Posted on
November 8, 2022
Updated on
February 1, 2023