1. ## Joint pdf Problem??

Let T1 and T2 be random times for a company to complete two steps in a certain process. Say T1 and T2 are measured in days and they have the joint p.d.f. that is uniform over the space 1 < t1 < 10, 2 < t2 < 6, t1 + 2t2 < 14. What is the P( T1 + T2 > 10)?
Thanks

2. Originally Posted by jojo_jojo
Let T1 and T2 be random times for a company to complete two steps in a certain process. Say T1 and T2 are measured in days and they have the joint p.d.f. that is uniform over the space 1 < t1 < 10, 2 < t2 < 6, t1 + 2t2 < 14. What is the P( T1 + T2 > 10)?
Thanks
The line T1+ 2T2= 14 is a diagonal of the rectangle with vertices (1,2), (10, 4), (10,6), and (1,6). The joint p.d.f is a constant, 1 over area of the triangle with vertices (1,2), (10,4), and (1,6).

Graph that region and the line T1+ T2= 10. That will cut the region above in two points, where T1+ T2= 10 intersects T1+ 2T2= 15, and where T1+ T2= 10 intersects T2= 2. P(T1+T2> 10) is the constant above multiplied by the area of the triangle with those two points and (10, 2) as vertices.

3. Thanks for the quick reply, but would you please explain your answer a little bit. I mean I understand how you got the 3 vertices, but couldn't understand how you got (10,4). Second, I also unable to understand last 2 lines.

5. Ya now it makes sense, would you please explain your this statement (I think here 15 is typo) also:

Graph that region and the line T1+ T2= 10. That will cut the region above in two points, where T1+ T2= 10 intersects T1+ 2T2= 15, and where T1+ T2= 10 intersects T2= 2. P(T1+T2> 10) is the constant above multiplied by the area of the triangle with those two points and (10, 2) as vertices.

Thanks

6. Did you draw a picture? This is why you studied geometry and the calculus of multiple variables. Don't let the opportunity get away!

The area of the entire region:

$\displaystyle \int_{1}^{2}6-2\;d(t1) + \int_{2}^{10}(7-\frac{t1}{2}) - 2\;d(t1)$

Now find the volume of that other piece.

Of course, since it's Uniform, it really masquerades as a Single Variable problem. Anything else and it's a different world.