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Chapter 7: Q.7.11 (page 353)

Consider n independent flips of a coin having probability p of landing on heads. Say that a changeover occurs whenever an outcome differs from the one preceding it. For instance, if n=5 and the outcome isHHTHT, then there are 3 changeovers. Find the expected number of changeovers. Hint: Express the number of changeovers as the sum of n−1 Bernoulli random variables.

Short Answer

Expert verified

The expected number of changeovers is(n-1)2p(1-p).

Step by step solution

01

Given Information

Let n independent flips of a coin have probability pof landing on heads.

02

Explanation

Define indicator random variables Ijthat marks if there was a changeover between jthand (j+1)st flip, j=1,...,n-1. Observe that Ij=1if and only if we have Head in jthflip and Tail in (j+1)st flip or if we have Tail in jthflip and Head in (j+1)st flip.

So, we have that

PIj=1=2p(1-p).
03

Explanation

Define xas the total number of changeovers. We have that X=∑j=1n-1Ijand using the linearity of the expectation, we have that

E(X)=∑jEIj=∑jPIj=1

=(n-1)·2p(1-p).

04

Final Answer

The expected number of changeovers is(n-1)2p(1-p).

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