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A box contains ten sealed envelopes numbered 1,..., 10. The rst ve contain no money, the next three each contain $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 a random sample), and you get the largest amount in any of the envelopes selected. If X1;X2; andX3 denote the amounts in the selected envelopes, the statistic of interest is M = the maximum of X1;X2, and X3. (a) Obtain the probability distribution of this statistic. (b) What is the expected value and variance of this statistics?

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mathecoach

Answer:

a)

P(X = 0) = 5/10

P(X = 5) = 3/10

P(X = 10) = 2/10

b)

E = 0·5/10 + 5·3/10 + 10·2/10 = 3.5

V = 0^2·5/10 + 5^2·3/10 + 10^2·2/10 - 3.5^2 = 15.25

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