Common probability distributions: two-item distribution, beta distribution, Dirichlet distribution

Knowledge Points: Bernoulli distribution, two-item distribution, polynomial distribution, prior probability, posterior probability, conjugate distribution, beta distribution, beta-two-item distribution, negative two-item distribution, Dirichlet distribution, gamma function, distribution

One, Bernoulli distribution (Bernouli distribution)

Also known as 0-1 distribution, refers to a randomized trial, with only two results. That is, the value of a random variable is only 0 and 1.
Recorded as: 0-1 distribution or B (1,p), where p indicates a probability that the result is positive or 1 at a time in the lab.

Probability calculation:

P(X=0)=p0 P(X=1) = p1

Expected calculation:

E(X)=0?P0+1?p1= p

The simplest example is to toss a coin to predict whether the result is positive or reverse.

two, two-item distribution (binomial distrubution)

Represents the results of the n-th-knoop experiment.

Recorded as:x~b (n,p), where n is the number of experiments, p indicates the probability that each of the results of the trial is 1, X indicates the number of successes in n experiments.
Probability calculation:

Expected calculation:

The example is to ask for several times to toss a coin and predict the number of positive results.

three, polynomial distribution (multinomial distribution)

The polynomial distribution is an extension of the two-item distribution, and in the case of polynomial distributions, there are n results for each experiment.
Probability calculation:

Expected calculation:

The simplest example is to toss a sieve several times, counting the number of times each face is thrown.

four, priori probability, posterior probability, conjugate distribution Prior and posterior probabilities:

The concept of transcendental probability and posterior probability is relative, and the probability of posterior examination is usually the probability of adding new information on the basis of prior probability, so it is also commonly called conditional probability. For example, 5 balls in the lottery, there are 2 prizes, now there are five people to smoke, nickname ranked in the third, the question of xiaoming to draw the prize probability is? At the beginning of nothing to know, of course, Xiaoming drew the probability of the prize p (X = 1) = 2/5. But when it comes to knowing that the first person has won the prize, Xiao Ming's probability of winning the prize will change, P (X = 1| Y1 = 1) = 1/4. Another example is the language model in natural language processing, which needs to calculate the probability of a word being produced by a language model.

Common probability distributions: two-item distribution, beta distribution, Dirichlet distribution