find the probability that Y1 + Y2 is less than 1

Suppose that the random variables Y1 and Y2 have a joint probability distribution function f(y1, y2) given by

f(y1,y2)= 0<=y1<=y2, y1+y2<=2 and f(y1,y2)=0 elsewhere

(A) What is the probability that Y1 + Y2 is less than 1?

for this question, i have hard time findthe region of my integration, it seems like i need to use separate integral to do this, if we do it as dy2dy1 order, then y2 goes from y1 to 1-y1, but then y2 go from 0 to to separate function, so do i need to do it twice?

Re: find the probability that Y1 + Y2 is less than 1

Quote:

Originally Posted by

**wopashui** Suppose that the random variables Y1 and Y2 have a joint probability distribution function f(y1, y2) given by

f(y1,y2)=

0<=y1<=y2, y1+y2<=2 and f(y1,y2)=0 elsewhere

(A) What is the probability that Y1 + Y2 is less than 1?

for this question, i have hard time findthe region of my integration, it seems like i need to use separate integral to do this, if we do it as dy2dy1 order, then y2 goes from y1 to 1-y1, but then y2 go from 0 to to separate function, so do i need to do it twice?

**Draw the region defined by the support.** You will need to use your knowledge of multi-variable calculus, in particular double integrals. Your textbook or class notes should have exmaples to follow.

Re: find the probability that Y1 + Y2 is less than 1

i have drawn the region, but y1 would go to something depends on y2 eventually, it will end up with a function of y2, but we need a number here

we are given the answer is 1/32, i know y2 goes from 0 to y1 to 1-y1, but y1 will goes from 0 to some 1-y2, and 0 to y2, which will end up with a function of y2

Re: find the probability that Y1 + Y2 is less than 1

Are you SURE you are considering the correct region.

Checking

Excellent. This indicates we might be on the right track.

Make two modifications to the integral and you are done.

Show us what you get.

Re: find the probability that Y1 + Y2 is less than 1

it should be 0 to 0.5 and y1 to 1-y1