Probability integration question

[a_man]

Homework Statement

Let X denote the diameter of an armored electric cable and Y denote the diameter of the ceramic mold that makes the cable. Both X and Y are scaled so that they range between 0 and 1 . Suppose that X and Y have the joint density

f(x,y) = 1/y, 0<x<y<1
f(x,y) = 0, elsewhere

Find P(X + Y > 1/2)

Homework Equations

The Attempt at a Solution

Do not I just need to integrate 1/y from x to 1 with dx then from there integrate one more time with from 0 to 1/2 with dy ?

I am really confused with this question because if we integrate 1/y we get ln (y) and if we integrate one more time, we get y *(ln (y) - 1) but we cannot have ln (0).

Some please help me. This question is bothering me for the whole day.
BTW: I am sorry that I do not know how to put the integration symbol yet.

 

[LeonhardEuler]

It would probably be helpful to draw a diagram so that you can see the area where you need to integrate. You are looking for X+Y>1/2. Graph the line X+Y=1/2. This is the boundary of the constraint, and on one side of the line it holds, on the other it fails. Also, the function is nonzero only for X<Y, so this is another constraint on the area of integration. If you draw a picture, it should be clear what region to integrate in and how to set the limits.

As far as your second point about ln(y) for y=0, remember that it is y*ln(y). The limit has a definite value. (hint: L'Hopital's rule)

And about drawing integrals, hit the quote button to see how I did this:
$$\int_{x=0}^{x=1}f(x)dx$$

 

[Mene]

Hello,

Thank you for your question. It seems like you are on the right track with your solution. However, you are correct in noting that there is an issue with integrating ln(y) when y=0.

One way to approach this problem is to break it into two cases: when y<1/2 and when y>1/2. When y<1/2, we can integrate from x to 1/2 with respect to y, and then from 0 to 1/2 with respect to x. This will give us the integral ln(1/2)-ln(x). When y>1/2, we can integrate from x to 1 with respect to y, and then from 0 to 1/2 with respect to x. This will give us the integral ln(y)-ln(x). We can then add these two cases together to get the final probability.

I hope this helps. Good luck with your homework!