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(Solution Download) 49. At time t = 0, there is one individual alive in a cer- tain population. A pure birth process...

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49.   At time t = 0, there is one individual alive in a cer- tain population. A pure birth process then unfolds as follows. The time until the rst birth is exponen- tially distributed with parameter l. After the rst

population distribution is normal? [Hint: Assuming normality, it can be shown that

E 31S2 2 2 4 = 1n + 12 s4/ 1n - 12

birth, there are two individuals alive. The time until

In  general,  it  is  dif  cult  to    nd   u?

to  minimize

the   rst gives birth again is exponential with param-

eter l, and similarly for the second individual. Therefore, the time until the next birth is the mini- mum of two exponential (l) variables, which is ex- ponential with parameter 2l. Similarly, once the second birth has occurred, there are three individu- als alive, so the time until the next birth is an expo- nential rv with parameter 3l, and so on (the memo- ryless 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 l. (Hint: The likelihood is a product of exponential terms.)

MSE1u? 2 ,  which  is  why  we  look  only  at  unbiased estimators and minimize V1u? 2 .]


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