Beta函数/Gamma函数/二项分布概率分布

最近在学习贝叶斯方面的内容，有一个例子涉及到二项分布，然后还和beta函数，gamma函数有些关系，因为不太熟悉，所以整理一下。

$\mathrm{B}$函数

wiki: https://en.wikipedia.org/wiki/Beta_distribution
In Bayesian inference, the beta distribution is the conjugate(coupled, connected, or related) prior probability distribution for the Bernoulli, binomial, negative binomial and geometric distributions.
For example, the beta distribution can be used in Bayesian analysis to describe initial knowledge concerning probability of success such as the probability that a space vehicle will successfully complete a specified mission. The beta distribution is a suitable model for the random behavior of percentages and proportions.

The probability density function (pdf) of the beta distribution, for 0 ≤ x ≤ 1, and shape parameters α, β > 0, is a power function of the variable x and of its reflection (1 − x) as follows:

where $\Gamma (z)$ is the gamma function. The beta function, $\mathrm{B}$, is a normalization constant to ensure that the total probability is 1. In the above equations $x$ is a realization—an observed value that actually occurred—of a random process $X$.

$\Gamma$函数

wiki: https://en.wikipedia.org/wiki/Gamma_function

In mathematics, the gamma function (represented by Γ, the capital Greek alphabet letter gamma) is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers. If n is a positive integer,

$\Gamma$=$(n-1)!$

The gamma function is defined for all complex numbers except the non-positive integers. For complex numbers with a positive real part, it is defined via a convergent improper integral:

$\Gamma$与$\mathrm{B}$函数的关系

推导参考: https://blog.youkuaiyun.com/xhf0374/article/details/53946146
结论：
$\Gamma (x)={\int }_0^{+\infty }{e}^{-t}{t}^{x-1}\mathrm{d}t$

$\mathrm{B}(x,y)={\int }_0^1{t}^{x-1}(1-t{)}^{y-1}\mathrm{d}t$

$\mathrm{B}(x,y)=\frac{\Gamma (x)\Gamma (y)}{\Gamma (x+y)}$