Probability_Theory

This blog just works as a formula stack.

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Formula

Law of Total Probability

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

THEN for any event $B$:
$$P(B)=\sum \ast {i=1}^{n}P(A\ast i)P(B|A_i)$$

Bayes’ Theorem

IF:{$A_i:i=1,2,3,\dots ,n$} 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 $A_k(k\in 1,2,3,\dots ,n)$:

$$P(A_k|B)=\frac{P(A_k)P(B|A_k)}{\sum _{i=1}^{n}P(A_i)P(B|A_i)}$$

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

IF the random variable $X$ follows the binomial distribution with and $p\in [0,1]$, 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 :

$$

Poisson Distribution $$

IF a discrete random variable X is said to have a Poisson distribution, with parameter $$, we write $$ or $$.

THEN it has a probability mass function given by :

$$

Continuous Uniform Distribution $$

IF the probability density function of the continuous uniform distribution $$ is :

$$

THEN we write $$, and the cumulative distribution function is :

$$

Exponential distribution $$

IF the probability density function of the continuous uniform distribution $$ and the rate parameter $$ is :

$$

THEN we write $$, and the cumulative distribution function is given by :

$$

Normal Distribution $$

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:

$$

THEN We write $$. The parameter $$ is the mean or expectation of the distribution (and also its median and mode), while the parameter $$ is its standard deviation. The variance of the distribution is $$.

Standard Normal Distribution $$

IF $$.

THEN when $$, we write $$. It is described by this probability density function:

$$

and the cumulative distribution function is given by :

$$

To Be Continued…