A box contains 10 sealed envelopes number1.....10. The first five contain no mon

Question

A box contains 10 sealed envelopes number1.....10. The first five contain no money, the next threecontain $5, and there is a $10 bill in each of the last two. A sample of size 3 is selected with replacement (so we have arandom sample) ,and you get the largest amount in any of theenvelopes selected. If X1, X2, X3 denote the amounts in the selected envelopes,the statistic of interest is M=max(X1,X2,.X3). Obtain the probability distribution ofthis statistic. A box contains 10 sealed envelopes number1.....10. The first five contain no money, the next threecontain $5, and there is a $10 bill in each of the last two. A sample of size 3 is selected with replacement (so we have arandom sample) ,and you get the largest amount in any of theenvelopes selected. If X1, X2, X3 denote the amounts in the selected envelopes,the statistic of interest is M=max(X1,X2,.X3). Obtain the probability distribution ofthis statistic.

Explanation / Answer

First, calculate the probability that M = 0, that is, none of theenvelopes contain money. 0.5*0.5*0.5 = 0.125 Next, calculate the probability that M = 10, that is, at least oneof the envelopes contain $10, which is 0.488. The probability that M = 5 is thus 1 - 0.125 - 0.488 = 0.387. The probability distribution is: M     Pr 0       0.125 5       0.387 10     0.488

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