# Thread: trouble with the bounds of integration

1. ## trouble with the bounds of integration

verify that this is a valid jdf.

f(y1, y2)= 6y1^2 y2 0<=y1<=y2, y1+y2<=2
0 elsewhere

I was trying to integrate y1 from 0 to y2
and y2 from 0 to 2

also i tried y1 from 0 to y2
and y2 from 0 to 1
all times 2

but still not confident with my answers. any help would be great

2. Originally Posted by batemanl
verify that this is a valid jdf.

f(y1, y2)= 6y1^2 y2 0<=y1<=y2, y1+y2<=2
0 elsewhere

I was trying to integrate y1 from 0 to y2
and y2 from 0 to 2

also i tried y1 from 0 to y2
and y2 from 0 to 1
all times 2

but still not confident with my answers. any help would be great
A simple sketch graph will show you that the double integral you need to calculate is

$\displaystyle \int_{y_1 = 0}^{y_1 = 1} \int^{y_2 = - y_1 + 2}_{y_2 = y_1} 6 y_1^2 y_2 \, dy_2 \, dy_1$,

where the integral terminals should be evident (especially in hindsight) from a sketch graph of the region over which the joint pdf is non-zero (the triangle bounded by the vertical $\displaystyle y_2$-axis and the lines $\displaystyle y_2 = y_1$ and $\displaystyle y_2 = -y_1 + 2$. Note that the two lines intersect at (1, 1)).