Probability distribution probability distributions

In the field of machine learning, probability distribution plays an important role in the understanding of data. Whether it is effective or noisy data, if you know the distribution of data, then in the data modeling process will be a great revelation. This paper summarizes several common probability distributions, such as the distribution of discrete random variables representing the Bernoulli distribution and the distribution of continuous random variables representing Gaussian distributions. For each distribution, it not only gives its probability density function, but also analyzes its expectation and variance as well as several main statistics. At present, the content of the article is relatively concise, follow-up and continue to improve.

This paper is mainly from three aspects: a function: Gamma function Six distributions: Bernoulli distribution, two-item distribution, polynomial distribution, beta distribution, Dirichlet distribution, Gaussian distribution a theory: conjugate transcendental a function: Gamma function

The gamma function is the generalization of the factorial on the real number, with the following formula:

Gamma (k) =∫∞0xk−1e−xdx,k∈ (0,∞) \gamma (k) = \int_0^\infty x^{k-1} e^{-x} \, DX, \quad K \in (0, \infty)

The Gamma function has a special nature, namely:

Gamma (N) = (n−1)! \gamma (n) = (n-1)!

six major distributions Bernoulli distribution

The Bernoulli distribution (Bernoulli distribution) is about the probability distribution of the Boolean variable xϵ{0,1} x \epsilon \lbrace 0,1 \rbrace, its continuous parameters μϵ[0,1] \mu \epsilon [0,1] represents the variable x= 1 probability of x=1. The probability distribution can be written as follows:

P (x|μ) =b (x|μ) =μx (1−μ) 1−x p (x | \mu) = B (x | \mu) = \MU ^{x} (1-\MU) ^{1-x}

For the Bernoulli distribution, its expectation and variance are as follows:

E (x) =μe (x) = \mu

var (x) =μ (1−μ) var (x) = \mu (1-\MU)

Two item distributions

The two-item distribution (binomial distribution) describes the probability of M-m successes (i.e. X=1 x=1) in N-N independent Bernoulli distributions, where the probability of success for each of the Knoop experiments is μϵ[0,1] \MU \epsilon [0,1].

P (m|n,μ) =bin (m|n,μ) =cmnμm (1−μ) N−m p (m|n,\mu) = Bin (M | n,\mu) = c_{n} ^{m} \mu^{m} (1-\MU) ^{n-m}

For the two-item distribution, it is the generalization of the Bernoulli distribution, and for the independent event, the mean value of the sums equals the sums of the mean, and the variance of sums equals the sums of the variance. So the expectation and variance are as follows: