Question

Brad and Chad meet at the local CrossFit gym and hit it off. However, since then they’ve only run in to each other a couple of times. Suppose Brad arrives at the gym (t=0), and that he always arrives before Chad. Let X be the duration of time Brad spends working out. On average, he spends 2 hours at the gym. Let Y be the amount of time until Chad arrives at the gym. On average, he arrives 1.5 hours after Brad arrives. Finally, assume that the time Brad spends at the gym is independent of the time of Chad’s arrival.

a) What distribution would you use to model X? What distribution would you use to model Y? Be sure to correctly specify the parameter(s) of the distribution.

b) What’s the probability that Brad spends 3 or more hours at the gym today?

c) What’s the probability that Brad leaves within 1.5 hours AND Chad arrives at least 2 hours after Brad’s arrival?

d) What’s the probability that Brad and Chad’s time at the gym overlaps?

Answer 1

Brad and Chad meet at the local CrossFit gym and hit it off. However, since then they’ve only run in to each other a couple of times. Suppose Brad arrives at the gym (t=0), and that he always arrives before Chad. Let X be the duration of time Brad spends working out. On average, he spends 2 hours at the gym. Let Y be the amount of time until Chad arrives at the gym. On average, he arrives 1.5 hours after Brad arrives. Finally, assume that the time Brad spends at the gym is independent of the time of Chad’s arrival.

• a) What distribution would you use to model X? What distribution would you use to model Y? Be sure to correctly specify the parameter(s) of the distribution.

We model both X and Y with exponential distribution because both describes the time between events in a Poisson point process (a process in which events occur continuously and independently at a constant average rate).

Given, E[X] = 2 hours and E[Y] = 1.5 hours.

Parameter of exponential distribution, = 1/mean

Thus, X ~ Exponential( = 1/2)

Y ~ Exponential( = 1/1.5)

• b) What’s the probability that Brad spends 3 or more hours at the gym today?

Probability that Brad spends 3 or more hours = P(X > 3) = exp(- * 3) = exp(-3/2) = 0.2231

• c) What’s the probability that Brad leaves within 1.5 hours AND Chad arrives at least 2 hours after Brad’s arrival?

Probability that Brad leaves within 1.5 hours = P(X < 1.5) = 1 - exp(- * 1.5) = 1 - exp(-1.5/2) = 0.5276

Probability that Chad arrives after 2 hours of Brad's arrival  = P(Y > 2) = exp(- * 2) = exp(-2/1.5) = 0.2636

Since X and Y are independent events,

Probability that Brad leaves within 1.5 hours and Probability that Chad arrives after 2 hours of Brad's arrival

=  0.5276 * 0.2636 = 0.1391

• d) What’s the probability that Brad and Chad’s time at the gym overlaps?

Probability that Bras and Chad's time overlaps = P(Y < X)

= (1/2) ( 2 - 3/3.5)

= 0.5714