Exercise 4.26

Answers

(a): Let $Z$ represent the number of flips until both Nick and Penny flip Heads. Then is $Z \sim FS (p_{1} p_{2})$ , since Nick’s and Penny’s flips are independent. $E (Z) = \frac{1}{p_{1} p_{2}}$
(b): The logic is analogous to part $a$ , but success probability is $p_{1} + p_{2} - p_{1} p_{2}$ .
(c): $P (X_{1} = X_{2}) = \sum_{k = 1}^{\infty} ({({(1 - p)}^{2})}^{k - 1} p^{2}) = \frac{p}{2 - p}$
(d): By symmetry, $P (X_{1} < X_{2}) = \frac{1 - (\frac{p}{2 - p})}{2} = \frac{1 - p}{2 - p}$

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2021-12-05 00:00