### Assignment Instructions/ Description

Image transcription textProblem 4. Flip a fair coin several times. Say that a changeover occurs whenever an outcome differs from

the one preceding it. For instance, if you flip the coin 5 times and the outcome is HHTHT, then there are 3

changeovers.

(i) Consider n independent flips of this coin. What is the expected number of changeovers?

Hint: For each flip, what is the probability that you observe a changeover? How can you decompose

the number of changeovers into a sum of bernoulli/indicator random variables?

(ii) If there are & changeovers, then the sequence of outcomes can be decomposed into & + 1 subsequences,

with each subsequence all heads or all tails, and the last subsequence of length 1. For example, the

following are outcomes with k = 2:

THT

HH TTT H

TTTTTTTTT HHHHHH T

What is the expected number of flips to get & changeovers?

Hint: How many flips are you doing before the next changeover? Can you model it as a known random

variable? Can you then express the number of flips you need to get & changeovers as a sum of known

random variables?... Show moreï¿½