Concrete mathematics : a foundation for computer science

8 EXERCISES 423
C What is the average total number of tickets you buy, if you continue
to play until going broke?
d What is the average number of games until you lose everything if
A doubledoubloon. you start with two doubloons instead of just one?
Bonus problems
52
53
54
55
56
57
Show that the text’s definitions of median and mode for random variables
correspond in some meaningful sense to the definitions of median and
mode for sequences, when the probability space is finite.
Prove or disprove: If X, Y, and Z are random variables with the property
that all three pairs (X, Y), (X, Z) and (Y, Z) are independent, then X + Y
is independent of Z.
Equation (8.20) proves that the average value of \iX is VX. What is the
variance of VX?
A normal deck of playing cards contains 52 cards, four each with face
values in the set {A,2,3,4,5,6,7,8,9,1O,J,Q,K}. Let X and Y denote
the respective face values of the top and bottom cards, and consider the
following algorithm for shuffling:
Sl Permute the deck randomly so that each arrangement occurs with
probability l/52!.
S2 If X # Y, flip a biased coin that comes up heads with probability p,
and go back to step Sl if heads turns up. Otherwise stop.
Each coin flip and each permutation is assumed to be independent of all
the other randomizations. What value of p will make X and Y independent
random variables after this procedure stops?
Generalize the frisbee problem of exercise 48 from a pentagon to an
n-gon. What are the mean and variance of the number of collision-free
throws in general, when the frisbees are initially at adjacent vertices?
Show that, if m is odd, the pgf for the number of throws can be written
as a product of coin-flipping distributions:
(m-1 l/2
Pk=
G,(z) = n ~
k=, 1 -qkz’
2 (2k- 1)~
where pk = sin
2m
Hint: Try the substitution z = l/cos2 0.
, qk = cos2 ‘2k2-; In.
Prove that the Penney-ante pattern ‘~1~2 . . . ~~~1~1 is always inferior to
the pattern jszrlr2 . . . ‘cl-1 when a fair coin is flipped, if 1 3 3.