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)= $\displaystyle 6y1^2y2 $ 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)= $\displaystyle 6y1^2y2 $ 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 $\displaystyle \int_{0}^{1}\int_{x_{1}}^{2-x_{1}}\;6\cdot x_{1}^{2}\cdot x_{2}\;\;dx_{2}\;dx_{1}\;=\;1$

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