Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 840 — #848
840
Chapter 19
Random Variables
B
H
A
T
C
H
T
H
T
C
B
Figure 19.10
Outcome Tree for Flipping Until HH or TT
Problem 19.16.
A coin with probability p of flipping Heads and probability q WWD 1 p of flipping
tails is repeatedly flipped until two consecutive flips match—that is, until HH or
TT occurs. The outcome tree A for this setup is illustrated in Figure 19.10.
Let e.T / be the expected number of flips starting at the root of subtree T of A.
So we’re interested in finding e.A/.
Write a small system of equations involving e.A/; e.B/, and e.C / that could be
solved to find e.A/. You do not need to solve the equations.
Homework Problems
Problem 19.17.
We are given a random vector of n distinct numbers. We then determine the maximum
of these numbers using the following procedure:
Pick the first number. Call it the current maximum. Go through the rest of the
vector (in order) and each time we come across a number (call it x) that exceeds
our current maximum, we update the current maximum with x.
What is the expected number of times we update the current maximum?
Hint: Let X i be the indicator variable for the event that the ith element in the
vector is larger than all the previous elements.