Two bankers each arrive at the station at some random time between 5PM
and 6PM (arrival time for each of them is uniformly distributed). They
stay exactly five minutes and then leave. What is the probability that
they will meet on a given day?
I am not sure how to go about modelling this problem as uniform distribution and solving it. Appreciate any help.
Here is how I start with it: Assume banker A arrives X minutes after 5PM and B arrives Y minutes after 5PM. Both X and Y are uniformly distributed between 5PM and 6PM. So pdf of X, Y is $\frac{1}{60}$. Now A and B will meet if $|X – Y| < 5$.
So required probability is $P(|X – Y| < 5)$ = Integral of joint distribution function of $|X – Y|$ from $0$ to $5$?
Now not sure how to write the equation from this point onwards and solve it.
Answer: $\frac {23}{144}$
Best Answer
Hint: If $f_{X,Y}$ is the multivariate pdf then you want to solve the following integral $$ \int_{0}^{60}\int_{\max\{0,x-5\}}^{\min\{60,x+5\}} f_{X,Y}(x,y)dydx. $$