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Exercise 3.2

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(a)

Since the trials are independent, the probability that the first

k - 1

trials fail is

{(\frac{1}{2})}^{k - 1}

, and the probability that the

k

-th trial is successful is

\frac{1}{2}

. Thus, for

k \geq 1

P (X = k) = {(\frac{1}{2})}^{k - 1} * \frac{1}{2} .

(b)

This problem reduces to part

a

once a trial is performed. Whatever it’s outcome, we label it failure and proceed to perform more trials until the opposite outcome is observed. Thus, for

k \geq 2

P (X = k) = {(\frac{1}{2})}^{k - 2} * \frac{1}{2} .

fifthist

2021-12-05 00:00

Exercise 3.2

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