Learning and finishing of probability theory 8: Geometric and hypergeometric distributions

1 Geometric distribution

1.1

Formulas for geometric distribution

E(X) = P*(1-p)^n-1

1.2

The reason for the geometric distribution

The first thing to understand is the geometric mean

(a*b)^-2

It is the proportional data

Each number is the geometric mean of the two numbers before and after

p*(1-p) ,p*(1-p)^2 ,p*(1-p)^3

p*(1-p)*p*(1-p)^3 =p^2*(1-p)^4

1.3 Expectation, variance of geometric distribution

2 hypergeometric distribution

2.1 Hypergeometric distribution

Prerequisites

No replacement sampling!

2.2 Definition of Hypergeometric Distribution

• It is called a hypergeometric distribution because its form is related to the coefficients of the series expansion of the "hypergeometric function".
• The parameters in the hypergeometric distribution are M,N,n, and the above hypergeometric distribution is denoted as X~H(n,M,N).
• f(k,n,K,N) = C(k,K) * C(n-k,N-K) / C(n,N)

• Understanding the formula
• f(k,n,K,N) = C(k,K) * C(n-k,N-K) / C(n,N)
• C(n,N) Number of ways to draw N samples in the population of n
• C(k,K) The number of methods for extracting K samples in all K designated special special units
• The number of combinations, the binomial coefficient
• C(n-k,N-K) Leave non-special n-k in n, the number of methods to extract N-k number

   
           
Sampling and replacement

2.3

• Both ends of the hypergeometric distribution
• If N=1, it degenerates to Bernoulli distribution.
  Indicates that it is only drawn once.
• 
  
• If the sample size n=1, that is, only one case is drawn from the finite population, and the probability that the case meets the requirements is just drawn, then the hypergeometric distribution can be restored to the binomial distribution.
• If only 1 sample is drawn at a time, no replacement sampling = replacement sampling, binomial distribution, quasi-normal distribution?
• If the capacity N of the data population is infinite, that is, the finite population is replaced by an infinite population. At this time, the replacement of the selected cases or not has no effect on the proportion of cases that meet the requirements in the population, and the hypergeometric distribution can also be used.regarded as a binomial distribution.
• N infinity, and no replacement sampling = replacement sampling, binomial distribution, quasi-normal distribution?
• In practical applications, as long as the number of cases in the data population is more than 10 times the sample size, that is, N>10n, the binomial distribution can be used to approximate the hypergeometric distribution, and the probabilities calculated by the two probability mass functions are almost the same