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Following the usual notation, derive a formula for the 3rd central moment in terms of the raw moments. The moment generating function (mgf) of the Negative Binomial distribution with parameters p and k is given as M(t)= p 1-(1-p)e' ¬k Use this mgf to derive general formulae for the mean and variance of the Negative Binomial distribution Show all the stens involved in your derivation.

Following the usual notation, derive a formula for the 3rd central moment in terms of the raw moments. The moment generating function (mgf) of the Negative Binomial distribution with parameters p and k is given as M(t)= p 1-(1-p)e' ¬k Use this mgf to derive general formulae for the mean and variance of the Negative Binomial distribution Show all the stens involved in your derivation.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.CR: Chapter 13 Review
Problem 6CR
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Define the rth central moment (ur) and rth raw moment (ar). Following the usual notation, derive a formula for the 3rd central moment in terms of the raw moments. The moment generating function (mgf) of the Negative Binomial distribution with parameters p and k is given as M(t) p 1-(1 - p)e' k Use this mgf to derive general formulae for the mean and variance of the Negative Binomial distribution. Show all the steps involved in your derivation.
Transcribed Image Text:Define the rth central moment (ur) and rth raw moment (ar). Following the usual notation, derive a formula for the 3rd central moment in terms of the raw moments. The moment generating function (mgf) of the Negative Binomial distribution with parameters p and k is given as M(t) p 1-(1 - p)e' k Use this mgf to derive general formulae for the mean and variance of the Negative Binomial distribution. Show all the steps involved in your derivation.
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Two independent random variables X1 and X2 follow Negative Binomial distributions with parameters (p, k₁) and (p, k2) respectively. Show, using the mgf given in part (b), that the sum X₁+X2 follows a Negative Binomial distribution with parameters p and (k₁+k2). Show your steps clearly giving reasons. The number of claims for a portfolio for a one week period has a Negative Binomial distribution with p=0.4 and k=6. Assuming claim numbers are independent from one week to another, calculate the probability of 9 claims in three weeks. Work out the expected number of claims in a five week period for this portfolio. Given that there were 45 claims in the first five weeks, calculate the probability that 20 of them were made in the first two weeks.
Transcribed Image Text:Two independent random variables X1 and X2 follow Negative Binomial distributions with parameters (p, k₁) and (p, k2) respectively. Show, using the mgf given in part (b), that the sum X₁+X2 follows a Negative Binomial distribution with parameters p and (k₁+k2). Show your steps clearly giving reasons. The number of claims for a portfolio for a one week period has a Negative Binomial distribution with p=0.4 and k=6. Assuming claim numbers are independent from one week to another, calculate the probability of 9 claims in three weeks. Work out the expected number of claims in a five week period for this portfolio. Given that there were 45 claims in the first five weeks, calculate the probability that 20 of them were made in the first two weeks.
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