Probability: joint probability distribution problem?

libelec

1. The problem statement, all variables and given/known data
John and George are set to meet each other at 12 o'clock. John's time of arrival, J, is distributed uniformly between 12:00 and 12:15. John will wait for George for 15 minutes. If he doesn't show up, he leaves. George's time of arrival, G, is also uniformly distributed, between 12:05 and 12:20. But he will only wait 5 minutes for John.

Find the probability of an encounter.

3. The attempt at a solution

I'm at complete lost with this problem. I think that what I have to do is find the jount probability distribution of J and G, but I couldn't say why. And the "waits for 15 minutes" thing also confuses me.

Any ideas?

libelec

Anybody?

vela

Hint: Let T=G-J. For what range of values of T will an encounter happen?

libelec

T between 0:00 and 0:05...

OK, and I have to do the same thing but considering T2 = J-G. How can I then find the entire answer? T+T2?

vela

Note that T2 = -T, so it's essentially the same variable. In other words, if you have something like a<T2<b, that's the same as -a>T>-b. You only have to work with one variable. There's no need to work the cases out separately and combine them at the end.

libelec

No, no, I see. T has to be between -0:15 and 0:05, right? Because if G<J, T<0, and since J can only be 15 minutes earlier than G otherwise he leaves, T> -0:15. and if G>J, T>0, and since G can only be 5 minutes earlier than J, T<0:05.

So, what I'm looking for is P(-0:15<= T <= 0:05), right?

vela

Right!

libelec

OK, thanks.

cgchayan

All understood.. but how to find out the p.d.f of T? Because without the pdf of T the required probability cannot be calculated..

Please help