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Negative Binomial

By Moloy De posted Thu March 18, 2021 08:46 PM

  
Suppose you are tossing a coin 10 times where the probability of getting a Head is p between 0 and 1. If you know the coin to be unbiased then consider p = 1/2. Let X denote the number of Heads. Then P(X = 6) = ( 10 choose 6) p6 (1-p)10-6 . So X is said to follow the Binomial Distribution with (10, p). The mean of distribution of X is p * 10 and variance is p * (1-p) * 10.

Now, suppose you are tossing the same with until Tail occurs. Let X denote the number of Heads till a Tail occurs. Then P(X = 6) = p6 (1-p) . X is said to follow Geometric Distribution.

Now, suppose you are tossing the same coin till 3 Tails occur. Let X denote the number of Heads till 3 Tails occur. Then P(X = 6) = (X+2 choose 2) p6+2 (1-p)3.  X is said to follow Negative Binomial Distribution with (3,p) which is a generalized version of Geometric Distribution. This distribution was first studied in 1713, by Montmort, as the distribution of the number of trials required in an experiment to obtain a given number of successes. It had previously been mentioned by Pascal.

Say for example Pat Collis is required to sell candy bars to raise money for the 6th grade field trip. There are thirty houses in the neighborhood, and Pat is not supposed to return home until five candy bars have been sold. So the child goes door to door, selling candy bars. At each house, there is a 0.6 probability of selling one candy bar and a 0.4 probability of selling nothing. What's the probability that Pat finishes on the tenth house? What's the probability that Pat finishes on or before reaching the eighth house? Etc.

Negative Binomial is commonly used to model Waiting Times, like stay in hospital. It is also commonly used to model data in the form of discrete sequence read counts from high-throughput RNA and DNA sequencing experiments.

QUESTION I: What are the mean and variance of Negative Binomial Distribution?
QUESTION II: Is the Negative Binomial Distribution positively Skewed? 

REFERENCE : Wikipedia

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