Question

In: Statistics and Probability

Lee and Martin compete in an arcade game where the two have to press a button...

Lee and Martin compete in an arcade game where the two have to press a button as soon as an alarm sound is triggered. Who presses the button first wins. Suppose the response times of Lee and Martin, X1 and X2, are independent random variables where Xi is exponentially distributed with rate λi.
a) What is the distribution of the time between the alarm sound and the first response?

b) What is the probability that the time until the first response exceeds T?

c) Suppose the game is played for n rounds and the response times of Lee and Martin are independent and identically distributed across different games. What is the variance of the number of rounds in which the response time exceeds T?

Solutions

Expert Solution

The response time of Lee is

the pdf is

The CDf is

The response time of Martin is

the pdf is

The CDf is

a) What is the distribution of the time between the alarm sound and the first response?

Let Z be the the time between the alarm sound and the first response. The first response is the minimum of response times of Lee and Martin

Hence

The cdf of Z is

The pdf of Z is

This is the pdf of an exponential distribution with rate

b) What is the probability that the time until the first response exceeds T?

the probability that the time until the first response exceeds T is

c) Suppose the game is played for n rounds and the response times of Lee and Martin are independent and identically distributed across different games. What is the variance of the number of rounds in which the response time exceeds T?

the probability that the time until the first response exceeds T on any given round is

Let Y be the number of rounds out of n, in which response time exceeds T. Since the response times of Lee and Martin are independent and identically distributed across different games, we can say that Y has a Binomial distribution with parameters, number of trials (number of rounds played) n=n and success probability (the probability that the time until the first response exceeds T on any given round) p=.

The Binomial probability that Y=y rounds in which response time exceeds T is

The variance of Y using the formula for Binomial variance is

ans: the variance of the number of rounds in which the response time exceeds T is

orchestra answered 2 years ago
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