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.

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

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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Step 1: Mention the given data in the question

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Step 2: Find the third central moment

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Step 3: Calculate mean and variance

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Follow-up Questions

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Follow-up Question

this is a follow up question

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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