Question

In: Statistics and Probability

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?

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 1 year ago

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