Beta distribution (probability of probability)

Catalog

1. Preface

2. Definition

 3.Beat The probability density function of the distribution （PDF）：

4.Beat Cumulative density function of distribution （CDF）：

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1. Preface

Bernoulli's test （ Repeatedly under the same conditions 、 A randomized trial conducted independently of each other , It is characterized by the fact that there are only two possible results of the randomized trial ： Happen or not ）

Frequency school Point of view （ The situation with the most occurrences reflects the distribution of probability ）, Embodies the Posttest

Gamma function ： Generalization of factorial in real number field .

2. Definition

For simple models like coin tossing or dice , We can define the probability distribution in advance . But in general , It is impossible to know the probability distribution of the system accurately . according to Frequency school Point of view , Sure Estimate the distribution of probability by frequency . For example, flip an uneven coin ,100 Next time 55 Sub head up , We can get the best estimate of the next result （ The probability of the coin appearing on the front is 55%）, But it's not entirely certain . therefore Probability is also a random variable , accord with Beat Distribution , The domain of definition is (0,1),Beat Distribution Generally used for modeling Probability distribution of the success probability of Bernoulli test event .Beta Distribution is a continuous probability density distribution , Expressed as x~Beta（a,b）, By two parameters a,b decision , It's called shape parameter .

 3.Beat The probability density function of the distribution （PDF）：

 

 

B（α,β） In order to Normalization , So that the integral is 1.B（α,β） That is, the area of the graph （ It's a relationship with α,β The related constants ）,（ It's all about , It's kind of similar softmax）.

Expectations are more important , According to the expectation, we can see the bias . Pictured above E（B（2,8）)=2/(2+8)=0.2

4.Beat Cumulative density function of distribution （CDF）：

mixup The weight of is generated here