X 0 1

P p 1-p

No matter how much the difference is, there are only two possibilities, either this result or that result. To put it bluntly, it is either success or failure.

The possible results of binomial distribution are uncertain, even endless.

A distribution table with binomial distribution list

x 0 1 2……n

1-p)^n…… 1-p)^(n- 1……p^n( 1-p)^0

That is to say, when n= 1, this special binomial distribution will become a two-point distribution.

In other words, the two-point distribution is a special binomial distribution.

Suppose there are m unqualified products in n products, that is, the unqualified rate p = m/n. N products are randomly selected for inspection and X products are found to be unqualified. It is known that the probability function of x is p (x = k) = c (k, m) * c (n-k, n-m)/c (n, n), and k = max.

It is mathematically proved that LIMC (k, m) * c (n- k, n-m)/c (m, n) = b (n, p) (binomial distribution) is not difficult. Therefore, in practical application, as long as n >；; = 10n, and the number of unqualified products can be approximately described by binomial distribution.

Hypergeometric Distribution-Academic Encyclopedia-HowNet Space

Let me give you an example: suppose a batch of products has 100 pieces, of which 10 pieces are defective.

So:

(1) What is the probability that a product is genuine? If there are only two possible results (genuine or defective), it can be summarized as two-point distribution.

(2) Sampling the returned samples for n times, and the distribution of the quality products appears. This is a binomial distribution. First, the number of genuine products that may appear in these n tests is 0 ~ n；; It is equivalent to doing n tests, each time distributed at two points. In other words, if you test here n times, the probability of each test being true is 0.9.

(3) What is the quantity distribution of defective products of M pieces (≤ 100) without return? This problem is hypergeometric distribution. Of course, at this time, we need to discuss which is bigger, m or 10, so as to confirm the possible values of the distribution, so I won't go into details here.

(4) Normal distribution is the most common distribution in nature. This distribution is determined by two parameters-mean and variance. It is directly or indirectly related to other distributions. For example, the binomial distribution in this problem, in fact, everyone has drawn n times, and the final results are different, which is caused by sampling error. However, if many people (n) do this experiment (each person draws n times)

(Orthotropic distribution is often associated with the limit distribution of other distributions, that is, n →∞; If n is finite, then the distribution of n is the most complicated, that is, four results)