Distribution and Poisson distribution of two items

Recently, we learned the course of genome assembly, in which two distributions and Poisson distributions were discussed when using Kmer to estimate the size of the genome, and the course gave a thorough understanding of their origins and relationships, and combined with specific examples, this article summarizes it.

Through this example will also be true to feel the magic of mathematics, the transformation of mathematical formula, wonderful proof, the most magical is its application, reminds me of a very famous but I have not read the book-"The Beauty of Mathematics"

Body: Relationship between two-item distribution and Poisson's distribution

Defined

Two distribution: P (x=k) =CNKPK (1-p) (n-k)

Toss a coin, assuming that the coin is uneven, the probability of throwing the positive is p, then in the experiment of N-Toss, there is a probability of k-th positive

Poisson distribution: P (x=k) =λke-λ/k!

The probability of the number of passengers coming to the bus station in unit time is K. Assuming that the average number of passengers arriving is λ

Relationship between two-item distribution and Poisson distribution

n Very large, p very small Poisson distribution can be used to approximate two distributions, at this time Λ=NP

A visual explanation of the relationship:

From the Poisson distribution. The unit time is divided into n equal, called N time window. Then the probability of a guest at a certain time window is λ/n. (explained later, in fact, this is not true) then we can map the Poisson distribution to the two-item distribution: In a certain time window the passenger corresponds to the front coin thrown; K-Guest corresponds to throw K-positive. Therefore, the Poisson distribution and the two-item distribution are approximate.

Distribution and Poisson distribution of two items