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In Exercise 5.62, we considered two individuals who each tossed a coin until the first head appears. Let Y1 and Y2 denote the number of times that persons A and B toss the coin, respectively. If heads occurs with probability p and tails occurs with probability q = 1 − p, it is reasonable to conclude that Y1 and Y2 are independent and that each has a geometric distribution with parameter p. Consider Y1 − Y2, the difference in the number of tosses required by the two individuals.
- a Find E(Y1), E(Y2), and E(Y1 − Y2).
- b Find E(Y12), E(Y22), and E(Y1Y2) (recall that Y1 and Y2 are independent).
- c Find E(Y1 − Y2)2 and V(Y1 − Y2).
- d Give an interval that will contain Y1 − Y2 with probability at least 8/9.
5.62 Suppose that the probability that a head appears when a coin is tossed is p and the probability that a tail occurs is q = 1 − p. Person A tosses the coin until the first head appears and stops. Person B does likewise. The results obtained by persons A and B are assumed to be independent. What is the probability that A and B stop on exactly the same number toss?
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