Beta distribution of gamma function with two-item distribution polynomial distribution

http://blog.csdn.net/shuimu12345678/article/details/30773929

0-1 Distribution:

In one experiment, either 0 or 1 distribution, called 0-1 distribution.

Two items distributed:

To do the N-1-p experiment, the probability of 1 for each experiment was p, the probability of the experiment being 0 was, the probability that there was K for 1,n-k Times was 0, two distribution B (N,p,k).

Two-Item distribution calculation:

B (n,p,k) =

In other words, to do the N-P1 experiment, the probability of each experiment is 1, the probability of 0 is P2, there is p1+p2=1; ask X1 for the experiment for 1,X2 experiments for 0, there is x1+x2=n, the probability of the event B (X1,X2,P1,P2) is how much?

B (X1,X2,P1,P2) =

Polynomial distribution:

To promote, consider if there are three possible, that is, Bernoulli toss coin test, the coin is thicker, it is possible to stand up, that may be positive, negative, stand up, the probability is p1,p2,p3, then after the N-Test, the front appears X1 times, Reverse x2 times, Stand up X3 times (to ensure x1+x2+ X3=n) What is the probability?

Can follow the above law, guess the formula is:

The formula is correct, and this is the expression of the distribution of the polynomial, which is shown in the following sense:

All lined with n! Case, then for each positive, inverse, and vertical sequence:

To stand at the opposite end of the coin ... Li Anti-

All contain the x1!*x2!x3! of the whole arrangement, so it is known that it is established.

Gamma function:

The gamma function is the extension of the factorial, and its expression is

It is said that the use of distribution points can be obtained (specific methods do not know):

So it's easy to get into the natural number field:

Beta functions:

The learning gamma function is prepared for learning the beta function, and the expression for the beta function is

Beta functions are prepared for the beta distribution, and the definition for the beta distribution is:

Beta distribution of gamma function with two-item distribution polynomial distribution