A box contains ten sealed envelopes numbered 1, . . . , 10. The
first five contain no money, the next three each contains $5, and
there is a $10 bill in each of the last two. A sample of size 3 is
selected with replacement, 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)Find the possible values ofthe statisticM.
b)Obtain the populationdistribution p(x) for the Xi and display
it in a pmf table.
c)Obtain the probability distribution p(m) of the statistic M
and display it in a pmf table.
d)As an alternative solutionto c), use a tree diagram and show
how to compute the probabilitiesdirectly from the tree.
e)Carry out a simulation experiment to compare the distributions
of M for various sample sizes. (Hint: Write a computer program to
generate the digits 0-9 from a discrete uniform distribution.
Assign a value of x = 0 to the digits 0-4, a value of x = 5 to
digits 5-7, and a value of x = 10 to digits 8 and 9. Generate
samples of increasing sizes, keeping the number of replications
constant, and compute M = max(X1,..., Xn) from each sample.)How
does thedistribution change as n increases