**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 until the th head appears, where each coin flip comes up heads independently with probability . Prove that this distribution is given by

for . (This is known as the *negative binomial* distribution.**)**

The formula comes naturally when we consider the probability exactly heads happen in a sequence of flips(the last th flip is heads).

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

Let be the number of flips until the th heads.

We can solve this exercise by using the definition formula of expectation:

,

but there is another way to work around it.

Let be the number of flips needed to get the th heads when we already had exactly heads. Clearly .

Each is a geometric random variable with parameter . So we have:

.

The linearity of expectations yields:

.

**Exercise 2.16: **Suppose we flip a coin times to obtain a sequence of flips . A *streak *of flips is a consecutive subsequence of flips that are all the same. For example, if , , and are all heads, there is a streak of length starting at the third flip.(If is also heads, then there is also a streak of length starting at the third flip.)

**(a)** Let be a power of . Show that the expected number of streaks of length is .

**(b) **Show that, for sufficiently large , the probability that there is no streak of length at least is less than .(Hint: Break the sequence of flips up into disjoint blocks of 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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