Sum of two independent uniform random variables

[Pixel08]

Hi,

http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter7.pdf (see page 8, sum of two independent random variables).

I don't understand why they had to go further into the limits, 1 < z < 2. Why do they have to do that? And also, where did they get it from?

Can someone explain why? I've been looking at it for several hours now! :(

 

[tiny-tim]

Hi Pixel08!

I don't understand why they had to go further into the limits, 1 < z < 2. Why do they have to do that?

In ∫ f(z-y) dy, the integrand is 0 unless 0 < y < 1 and 0 < z - y < 1

The second condition is the same as y < z and y > z - 1

If z < 1, we can ignore the last condition (because z - 1 < 0), so that's 0 < y < 1 and y < z, ie 0 < y < z

If z > 1, we can ignore y < z (because y < 1 anyway), so that's 0 < y < 1 and y > z - 1, ie z - 1 < y < 1