Question

58. Each round played by a contestant is either a successwith probability p or a failure...

58. Each round played by a contestant is either a successwith probability p or a failure

with probability 1− p. If the round is a success, then a random amount of money

having an exponential distribution with rate λ is won. If the round is a failure,

then the contestant loses everything that had been accumulated up to that time

and cannot play any additional rounds. After a successful round, the contestant

can either elect to quit playing and keep whatever has already been won or can

elect to play another round. Suppose that a newly starting contestant plans on

continuing to play until either her total winnings exceeds t or a failure occurs.

(a) What is the distribution of N, equal to the number of successful rounds that

it would take until her fortune exceeds t?

(b) What is the probability the contestant will be successful in reaching a fortune

of at least t?

(c) Given the contestant is successful, what is her expected winnings?

(d) What is the expected value of the contestant’s winnings?

Math   Statistics-And-Probability   
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