Suppose you flip a biased coin (that lands heads with probability p) until 2 heads appear....

Suppose you flip a biased coin (that lands heads with probability p) until 2 heads appear. Let X be the number of flips needed for this two happen. Let Y be the number of flips needed for the first head to appear. Find a general expression for the condition probability mass function pY |X(i|n) when n ≥ 2. Interpret your answer, i.e., if the number of flips required for 2 heads to appear is n, what can you say about the arrival of the first head?

Solutions

Expert Solution

WE have has negative binomial distribution and has geometric distribution with parameter .

The PMF of is

The joint PMF of is found as follows: Observe that

has geometric distribution with parameter .

Thus,

Using the conditional probability rule,

In the notation given in the question,

If the second head appears in the n-the toss, then the first head appears in one of the tosses .

That is the second toss has equal probability of occurring in one of n-1 tosses independent of p. Hence the conditional probability

orchestra answered 11 months ago

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