For determining half-lives of radioactive isotopes, it is important to know what

Question

For determining half-lives of radioactive isotopes, it is important to know what the background radiation is in a given detector over a specific period. The following data were taken in a -ray detection experiment over 98 ten-second intervals:

58 50 57 58 64 63 54 64 59 41 43 56 60 50 46 59 54 60 59 60 67 52 65 63 55 61 68 58 63 36 42 54 58 54 40 60 64 56 61 51 48 50 60 42 62 67 58 49 66 58 57 59 52 54 53 53 57 43 73 65 45 43 57 55 73 62 68 55 51 55 53 68 58 53 51 73 44 50 53 62 58 47 63 59 59 56 60 59 50 52 62 51 66 51 56 53 59 57

Assume that these data are observations of a Poisson random variable with mean .

(a) Find the values of x and s2.

(b) What is the value of the maximum likelihood estima-

tor of ?

(c) Is S^2 an unbiased estimator of ?

(d) Which of x and s^2 would you recommend for estimat- ing ? Why? You could compare the variance of X with the variance of S2, which is

Var(S2)= [(2n+n1)]/[n(n1)]

Explanation / Answer

answer of part a) x-=56.29 and s2=56.12

ansewer of part b) maximum liklihood estimator of lambda is sample mean which is x-=56.29

answer of part c) mean and variance of poison is lambda so s2 will be also the unbiased estimator of lambda

answer var(s2)=[(2n+n1)]/[n(n1)]=65.91

var(x-)=var(x)/n=lambda/n=56.29/98=0.57

so x- should be used to estimate lambda as its variance is less than the variance of s2

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