Several important probability distributions (I)

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There are several important probability distributions:

Two-term distribution and Poisson distribution, Even distribution, exponential distribution, and positive distribution.

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1. benuli profile and two-item distribution

1. benoli Profile

N This repeat test performed under the same conditions, if each test has only two basic events that are relatively established, and their probability of occurrence in each test remains unchanged, so this test is called N-weight benuli test or benuli model.

For example, throwing a coin n times (front or back)

Play basketball n times (medium or medium)

Check n products (qualified or unqualified)

Set the probability of event a occurring in each test to P. (0 <p <1), the probability that event a happens exactly to m times in the benuli model is

M = 0, 1, 2 ,......, N; q = 1-P

Proof: we can see from the concept that multiple events are independent of each other that the probability of event a occurring m times specified in N tests and not occurring N-m times is pmqn-M, because there are CNM methods for extracting m from N tests, we can prove that.

2. Two-item distribution

Defines the probability distribution of random variable X

Where 0 <p <1, q = 1-P, I =, 2,..., n, is called discrete random variable X obeying two distributions of parameters N and P. X ~ B (n, p ).

Mathematical expectation of the two distributions E (x) = NP, variance d (x) = npq.

Is a two-item distribution of N = 20, P = 0.125:

Ii. Poisson distribution

Define all possible values of variable X as 0, 1, 2,... and the probability distribution is

And I = 0, 1, 2,...; λ is a constant, and λ> 0. X follows the Poisson distribution of λ and is counted as X ~ P (λ ).

Relationship between two-term distribution and Poisson distribution

(Poisson theorem)

Assume that random variable X is subject to two distributions B (n, p). When n → + ∞, X is approximately subject to Poisson distribution P (λ), that is

Here, λ = NP.

[PS: only when the value of P is small, generally less than 0.1, the error generated by replacing the two distributions with Poisson distribution is relatively small]

Mathematical expectation of Poisson distribution E (x) = λ, variance d (x) = λ.

This section compares a Poisson distribution with two distributions:

Let's look at the situation when P <0.1

The two are close to each other.