Math 331, Section 201—Homework #5
due in class Tuesday, March 25, 2003
Warm-Up Questions—do not hand in
Graham, Knuth, and Patashnik, Chapter 8, pp. 427–429, #1, #2, #3, #7, #11, #12(a),
#14, #24, #25
Homework Questions—turn in solutions to these problems
I. Graham, Knuth, and Patashnik, Chapter 8, p. 431, #37
II. (a) Find three random variables X, Y, and Z (on the same probability space) such
that X and Y are independent, Y and Z are independent, and X and Z are independent, but the triple X, Y, Z is not independent, that is, Pr( X = x and Y =
y and Z = z) is not always equal to Pr( X = x) Pr(Y = y) Pr( Z = z).
(b) Let X, Y, Z each denote the outcome of a fair 6-sided die (independent rolls).
(Call this uniform probability space of triples “3D”.) Are the random variables
X + Y and Y + Z independent?
(c) Consider the random variable S = X + Y on the uniform probability space
2D. Suppose Z is another random variable on 2D that is independent of S.
Prove that Z is a constant, that is, there is a value z such that Pr( Z = z) = 1.
III. Recall the example in class of the two “weird” 6-sided dice whose joint distribution
was identical to a regular pair of 6-sided dice (refer to problem 36(a) on page 431
of Graham, Knuth, and Patashnik). Find three different pairs of “weird” 8-sided
dice in a similar fashion (all the numbers on the sides of the dice are to be positive
integers). Which pair of dice should you use if you don’t like rolling doubles?
IV. A biased coin comes up heads with probability p and tails with probability 1 − p.
Find a value of p such that, using multiple independent flips of this coin, we can
find one event that has probability 12 and a second event that has probability 31 .
(Hint: consider n flips where n is very small, and consider the events “not all n
flips have the same outcome”.)
V. This problem concerns n-sided dice with the faces labeled 1, 2, . . . , n − 1, and ?.
(a) You play a solitaire game with the n-sided die described above. Your score is
the number rolled, in which case you stop; but if you roll a ?, you roll again
and score double the resulting number. This doubling can continue, so if you
roll ?, ?, ?, 7, then your score is 56. Calculate the expected value of your score.
(b) You modify the rules so that if you roll a ?, you next roll two of these n-sided
dice and add the scores together (without doubling). Each new n-sided die
might spawn new pairs as well, so a game could be ?, then ? and 5, then 2 and
9 (from the second star) for a total score of 16. Calculate the expected value of
your score under these rules.
(c) Restricting now to the case n = 6, calculate the variance of your score under
the rules in part (a); under the rules in part (b).
(d) Using Chebyshev’s inequality, find a number S so that you have at least a 99%
chance of scoring at most S under the first rules; under the second rules.