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At time t = 0, there is one individual alive in a certain population. A pure birth process then unfolds as follows. The time until the first birth is exponentially distributed with parameter λ. After the first birth, there are two individuals alive. The time until the first gives birth again is exponential with parameter λ, and similarly for the second individual. Therefore, the time until the next birth is the minimum of two exponential (λ) variables, which is exponential with parameter 2λ. Similarly, once the second birth has occurred, there are three individuals alive, so the time until the next birth is an exponential rv with parameter 3λ, and so on (the memoryless property of the exponential distribution is being used here). Suppose the process is observed until the sixth birth has occurred and the successive birth times are 25.2,41.7,51.2,55.5, 59.5, 61.8 (from which you should calculate the times between successive births). Derive the mle of λ. [ Hint: The likelihood is a product of exponential terms.]

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Probability and Statistics for Eng...

9th Edition
Jay L. Devore
Publisher: Cengage Learning
ISBN: 9781305251809
BuyFind

Probability and Statistics for Eng...

9th Edition
Jay L. Devore
Publisher: Cengage Learning
ISBN: 9781305251809

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Chapter
Section
Chapter 6, Problem 33SE
Textbook Problem

At time t = 0, there is one individual alive in a certain population. A pure birth process then unfolds as follows. The time until the first birth is exponentially distributed with parameter λ. After the first birth, there are two individuals alive. The time until the first gives birth again is exponential with parameter λ, and similarly for the second individual. Therefore, the time until the next birth is the minimum of two exponential (λ) variables, which is exponential with parameter 2λ. Similarly, once the second birth has occurred, there are three individuals alive, so the time until the next birth is an exponential rv with parameter 3λ, and so on (the memoryless property of the exponential distribution is being used here). Suppose the process is observed until the sixth birth has occurred and the successive birth times are 25.2,41.7,51.2,55.5, 59.5, 61.8 (from which you should calculate the times between successive births). Derive the mle of λ. [Hint: The likelihood is a product of exponential terms.]

Expert Solution
To determine

Derive the mle of λ.

Explanation of Solution

Given info:

A pure birth process gives the successive birth times as 25.2, 41.7, 51.2, 55.5, 59.5, 61.8. The process has an exponential distribution with parameter λ and is observed till the sixth birth.

Calculation:

Denote x1 as the time until the first birth. Denote xi as the time between the (i1)th and ith birth, where i=1,2,...,6.

It is known that, for a sample of size n, taken from a population with parameter λ, the likelihood function is given as:

f(x1,x2,...,xn;λ)=i=1nf(xi;λ)

Here, each random variable has an exponential distribution. Thus, the likelihood function is:

f(x1,x2,...,xn;λ)=i=1n(iλ)eiλxi=n!λneλi=1nixi.

Taking logarithm on both sides, the log likelihood function is:

ln(f)=ln(n!λneλi=1nixi)=ln(n!)+nln(λ)λi=1nixi

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Chapter 6 Solutions

Probability and Statistics for Engineering and the Sciences
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