A new intern was asked to estimate a probability distribution for the number of

Question

A new intern was asked to estimate a probability distribution for the number of items purchased by customers entering one o the company’s retail stores. The intern analyzed the data and decided that the negative binomial distribution was the appropriate probability distribution to use. The intern found that the parameters satisfy r = = q/p. The intern’s mentor reviewed the analysis and pointed out that the r parameter appeared to be correct but is wrong. The intern’s model indicates that 1/9 of customers make no purchases but in fact 4/9 of customers make no purchases. Calculate the variance of the corrected negative binomial distribution

Explanation / Answer

let x be the number of customers required to get rth per cheese

so here probability of customer purchase p=4/9

then q= 1-p = 1-4/9=5/9

r is caluclated by r=q/p =(1-p)/p =(5/9)/(4/9) =5/4

for negative binomial the variance is var(x)= rq/p^2 = (5/4*5/9)/(4/9)^2 = 3.515

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