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Suppose that in Problem 70 we continue to flip the

Suppose that in Problem 70 we continue to flip the coin until a head appears. Let N denote the number of flips needed. Find

(a) P{N ≥ i}, i ≥ 0;

(b) P{N = i};

(c) E[N].

Problem 70

Consider an urn containing a large number of coins, and suppose that each of the coins has some probability p of turning up heads when it is flipped. However, this value of p varies from coin to coin. Suppose that the composition of the urn is such that if a coin is selected at random from it, then the p-value of the coin can be regarded as being the value of a random variable that is uniformly distributed over [0, 1]. If a coin is selected at random from the urn and flipped twice, compute the probability that

(a) The first flip results in a head;

(b) Both flips result in heads.

(a) P{N ≥ i}, i ≥ 0;

(b) P{N = i};

(c) E[N].

Problem 70

Consider an urn containing a large number of coins, and suppose that each of the coins has some probability p of turning up heads when it is flipped. However, this value of p varies from coin to coin. Suppose that the composition of the urn is such that if a coin is selected at random from it, then the p-value of the coin can be regarded as being the value of a random variable that is uniformly distributed over [0, 1]. If a coin is selected at random from the urn and flipped twice, compute the probability that

(a) The first flip results in a head;

(b) Both flips result in heads.

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