Probability and Computing: Chapter 2 Exercises (Cont. 1)

Exercise 2.14: The geometric distribution arise as the distribution of the number of times we flip a coin until it comes up heads. Consider now the distribution of the number of flips X until the kth head appears, where each coin flip comes up heads independently with probability p. Prove that this distribution is given by

Pr(X=n) = \binom{n-1}{k-1}p^{k}(1-p)^{n-k}

for n \geq k. (This is known as the negative binomial distribution.)

The formula comes naturally when we consider the probability exactly k-1 heads happen in a sequence of n-1 flips(the last nth flip is heads).

Exercise 2.15: For a coin that comes up with head independently with probability p on each flip, what is the expected number of flips until the kth heads?

Let X be the number of flips until the kth heads.
We can solve this exercise by using the definition formula of expectation:

E[X]=\sum_{i=1}^{\infty}iPr(X=i) = \sum_{i=1}^{\infty}i\binom{i-1}{k-1}p^{k}(1-p)^{i-k},

but there is another way to work around it.
Let X_{i} be the number of flips needed to get the ith heads when we already had exactly i-1 heads. Clearly X=\sum_{i=1}^{k}X_{i}.
Each X_{i} is a geometric random variable with parameter p. So we have:
E[X_{i}]=\frac{1}{p}.
The linearity of expectations yields:
E[X]=E[\sum_{i=1}^{k}X_{i}]=\sum_{i=1}^{k}E[X_{i}]=\sum_{i=1}^{k}\frac{1}{p}=\frac{k}{p}.

Exercise 2.16: Suppose we flip a coin n times to obtain a sequence of flips X_{1},X_{2},\ldots ,X_{n}. A streak of flips is a consecutive subsequence of flips that are all the same. For example, if X_{3}, X_{4}, and X_{5} are all heads, there is a streak of length 3 starting at the third flip.(If X_{6} is also heads, then there is also a streak of length 4 starting at the third flip.)

(a) Let n be a power of 2. Show that the expected number of streaks of length log_{2}n+1 is 1-o(1).

(b) Show that, for sufficiently large n, the probability that there is no streak of length at least \lfloor log_{2}n-2log_{2}log_{2}n \rfloor is less than 1/n.(Hint: Break the sequence of flips up into disjoint blocks of \lfloor log_{2}n-2log_{2}log_{2}n\rfloor consecutive flips, and use that the event that one block is a streak is independent of the even that any other block is a streak.)

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