Probability of Xavier & Yolanda Meeting at Cafe

[simmonj7]

Homework Statement

Xavier and Yolanda both have classes that end at noon and they agree to meet every day after class. They arrive at a campus cafe independently. Xavier’s arrival time is X and Yolanda’s arrival time is Y , where X and Y are measured in minutes after noon. The
individual density functions are:
f1(x) = {e^−x if x ≥ 0,
0 if x < 0}
f2(y) ={y/50 if 0 ≤ y≤ 10,
0 otherwise.}
After Yolanda arrives, she will wait at the cafe up to half an hour for Xavier and then go to the library. On the other hand, if Xavier arrives and does not find Yolanda, he will email her a message and leave immediately for the library. Find the probability that they meet at the cafe.

The Attempt at a Solution

So I thought that you would just multiple f1(x) by f2(x) and then integrate that function to get the answer. However, I am not entirely certain of what the bounds would be for that integral.

Help please.
Thanks.

 

[LCKurtz]

1. Write down the inequalities that Y and X must satisfy for the cafe meeting to happen
2. Draw a picture in the xy plane showing the region that describes.

That picture is where you will get the correct limits.

 

[simmonj7]

I don't understand what these inequalities are or what this region that you are saying I should draw is.
I thought there was supposed to just be one function that was f1(x) times f2(y)...Or that is what I was lead to believe in class.

 

[LCKurtz]

Well, for example, if Xavier arrives he won't wait for Yolanda. What does that, by itself, tell you about the arrival times Y and X for successful meeting?

 

[simmonj7]

That they will only have a successful meeting if she arrives on time. But I don't get how that becomes bounds of integration.

 

[LCKurtz]

Well, for example, if Xavier arrives he won't wait for Yolanda. What does that, by itself, tell you about the arrival times Y and X for successful meeting?

That they will only have a successful meeting if she arrives on time. But I don't get how that becomes bounds of integration.

To say she "arrives on time" doesn't mean anything. The point is that the fact Xavier won't wait for her tells you a relation between their arrival times for a successful meeting. So you need to express that fact with an inequality between Y and X. And the fact that she won't wait more than 30 minutes tells you another relation which you can express with another inequality. You have to figure out those two inequalities and write them down with X's and Y's before you have a chance to figure out the limits. An English sentence description won't do; you need the mathematical statements.

 

[simmonj7]

Yes, I understand that a sentence in English won't do me any good.
However, what I have been trying to communicate with you is that I have no idea what to do on this problem and where to get these inequalities from or how to construct them. I have not done or seen a problem of this sort before and I am essentially clueless so what you are saying is not really helping me understand that.

 

[LCKurtz]

I know, I know. One thing at a time. Can you answer these two questions by reading the description of the problem:

1. If X < Y will they have a successful meeting?

2. If X > Y will they have a successful meeting?

 

[simmonj7]

Sorry about that I just really wasn't given a good background of this concept and want to fully understand it. Thank you.

But anyways, X is Xavier's arrival time and Y is Yolanda's arrival time.
So if X < Y, then Xavier arrives first and since he doesn't wait they will not have a successful arrival time.

If X > Y, Yolanda arrived first so they will have a successful arrival time as long as he arrives within 30 minutes after she arrived.

 

[LCKurtz]

Sorry about that I just really wasn't given a good background of this concept and want to fully understand it. Thank you.

But anyways, X is Xavier's arrival time and Y is Yolanda's arrival time.
So if X < Y, then Xavier arrives first and since he doesn't wait they will not have a successful arrival time.

Yes. So for success you can't have X < Y so you must have Y ≤ X. That is your first condition

If X > Y, Yolanda arrived first so they will have a successful arrival time as long as he arrives within 30 minutes after she arrived.

Yes. So X ≤ ? for success. You have to have both inequalities work for success.

Once you have these two inequalities, they will describe a region in the xy plane showing when the meeting happens. You also have the region in the xy plane where your joint density function that you have:

f(x,y) = f_X(x)f_Y(y)

is non-zero. The region in the xy plane where you have success is the region you must integrate your joint density function over. Starting to make sense?

 

[simmonj7]

That makes a lot of sense. Thank you very much.