A box contains ten sealed envelopes numbered 1, . .. , 10. The first four contai

Question

A box contains ten sealed envelopes numbered 1, . .. , 10. The first four contain no money, the next four each contains $5, and there is a $10 bil 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, and X3 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. (Round your answers to three decimal places.) 0 n(rm) b) Describe how you would carry out a simulation experiment to compare the distributions of M for various sample sizes. This answer has not been graded yet How would you guess the distribution would change as n increases? O As n, increases, p(M 0) goes to zero, p(M-5) goes to one. O As n, increases, p(M = 0) goes to one, p(M = 10) goes to zero. O As n increases, p(M 0) goes to zero, p(M-10) goes to one O As n, increases, p(M = 0) goes to zero, p(M = 10) goes to zero.

Explanation / Answer

(a) Possible values of M are 0,5,10

M=0 when all 3 envelopes contains 0 money,
hence P(M=0)=(0.5)3
                      =0.125

M=10 when there is at least one envelope with $10
hence P(M=10)=1-P(no envelope with $10)
                       =1-(0.8)3
                       =0.488

P(M=5)=1-(0.125+0.488)
            =0.387
The sampling distrubution of M is

As n,increases ,P(M=0) goes to zero ,P(M=10) goes to One

m 0 5 10 p(m) 0.125 0.387 0.488

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