Commonly used probability distributions: Bernoulli distribution, binomial distribution, polynomial distribution, Gaussian distribution, exponential distribution, Laplace distribution and Dirac delta d

Bernoulli distribution （Bernoulli distribution）

** Bernoulli distribution ：** Distribution of single binary random variable . By a single parameter φ∈[0,1] control .

example ： Flip a coin , The probability of facing up .

Binomial distribution (binomial distrubution)

Binomial distribution ： stay n Events in this test A It happens k The probability of time .
Probability calculation ：

In style k=0,1,2,…,n,

Expect to calculate ：E(x)=np

Variance calculation ：D(x)=np(1-p)

example ： throw n The second coin , Face up m The probability of time .

Polynomial distribution （multinoulli distribution）

Polynomial distribution ： have k Distribution on a single discrete random variable in different states , among k It's a finite value . Expansion of binomial distribution .

Probability calculation ：

example ： Multiple screen throwing , Count the probability of the number of times each side is thrown .

Gaussian distribution （Gaussian dis-tribution）

Gaussian distribution （Gaussian dis-tribution）, Also known as normal distribution （normal distribution）. namely ：


notes ： The normal distribution presents a classic “ A bell curve ” The shape of the , Of which the central peak x Coordinates from µ give , The width of the peak is affected by σ control . In this example , What we're showing is Standard normal distribution （standard normaldistribution）, among µ=0,σ=1.
Expect to calculate ：E(x)=µ
Variance calculation ：D(x)=σ^2

An index distribution （exponential distribution）

An index distribution （exponential distribution）： stay x=0 Get the boundary point （sharp point） The distribution of . namely ：

Indicator function （indicator function）：
The exponential distribution is made proper by the indicator function x The probability of taking a negative value is zero .

Laplacian distribution （Laplace distribution）

Laplacian distribution （Laplace distribution）： Probability density function distribution of random variables . namely ：

Dirac Distribution

Dirac delta function （Dirac delta function）： Defined as in addition to 0 Values for all points except 0, But the integral is 1.

Dirac Distribution is often used as an empirical distribution （empirical distribution） A component of ：