Summer 2009 CMSC 351: Homework 7 Clyde Kruskal
Due at the start of class, Friday, June 26.
Problem 1. For this problem you may use a calculator for a few calculations.
Consider P100
k=1 k3/2.
(a) Use a non-integral method to show that the sum is between 15,000 and 70,000.
(b) Approximate the sum using integrals. Make sure to get an upper and lower
bound.
Problem 2. A carnival game consists of three dice in a cage. A player can bet a dollar on
any of the numbers 1 through 6. The cage is shaken, and the payoff is as follows. If
the playerās number doesnāt appear on any of the dice, he loses his dollar. Otherwise
if his number appears on exactly kof the three dice, for k= 1,2,3, he keeps his dollar
and wins kmore dollars.
(a) What is his expected gain from playing the carnival game once?
(b) What is the variance from playing the carnival game once?
(c) What is the standard deviation from playing the carnival game once?
(d) What are his expected gain, variance, and standard deviation from playing the
carnival game ntimes?
Problem 3. Consider the bubble-sort algorithm, for an input (a b c) of the three integer
values 1, 2, and 3 in any order, where all input orders are equally likely (uniform
distribution).
(a) Let Mbe the number of exchanges. What are E(M), Var(M), and Ļ(M)?
(b) Let xbe the number of comparisons. What are E(M), Var(M), and Ļ(M)?
(c) What p ercentage of the sample space is within a standard deviation of average.
Problem 4. Again consider the bubble-sort algorithm, for an input (a b c) of the three
integer values 1, 2, and 3 in any order. But now suppose the input orders are not
equally likely; instead, there is a 1/5 chance of a= 1 being the first item in the input
(the 1/5 distributed equally among those inputs) and 4/5 of anot being 1 (again the
4/5 shared equally among all these inputs).
(a) Let Mbe the number of exchanges. What are E(M), Var(M), and Ļ(M)?
(b) Let xbe the number of data comparisons made. What are E(M), Var(M), and
Ļ(M)?
(c) What p ercentage of the sample space is within a standard deviation of average.