# Thread: Much needed help on this integral/density function/probabilty world problem

1. ## Much needed help on this integral/density function/probabilty world problem

Question:

Fatima and Zack both have labs which end at noon. Every day after the lab they arrive at Satrbucks
located on campus, independently. Zack arrives at time X and Fatima’s arrival time is Y, where X and Y are
measuared in mintues afternoon. The indiviual density functions are:

f1(x) =

e^-x if x is greater or equal to 0,
0 if x < 0

f2(y) =

(1/50)*y if 0 is less than or equal to y less than or equal to 10,
0 if otherwise.

(Zach arrives sometime after noon and is more likely to arrive promptly than late. Fatima always arrives by
12:10 PM and is more likely to arrive late than promptly.) After Fatima arrives, she’ll wait for up to half an hour for
Zack, but he won’t wait for her. Find the probability that they meet.

Thanks!

2. ## Re: Much needed help on this integral/density function/probabilty world problem

Simplest would be to get the density function for Zach arriving and bothering to stick around because Fatima is already there. At any moment between x=0 and x=10 the probability of that is e^(-x) (Zach arriving) times x^2/100 (Fatima already there). During the following 20 minutes, the second of these is one, instead. So evaluate the two definite integrals.

Or form double integrals, e.g.,

$\int_0^{10} \int_0^x e^{-x}\ \frac{y}{50}\ dy\ dx$

... where Fatima arriving first is the region under the line y=x.