Homework Help: Bivariate probabilities

1. The problem statement, all variables and given/known data
The problem statement, all variables and given/known data[/b]
Let X and Y have the joint pdf
fXY(x,y) = x + y, 0<x<1, 0<y<1

determine Pr X+Y , 1/2


2. Relevant equations



3. The attempt at a solution
[ tex ]
\int_{0}^{\frac{1}{2}}\int_{0}^{\frac{1}{2}}x+y dx dy

[ /tex ]
= 0.125
1. The problem statement, all variables and given/known data



2. Relevant equations



3. The attempt at a solution

 

Sorry I didn't put the code in right

[itex]
\int_{0}^{\frac{1}{2}}\int_{0}^{\frac{1}{2}}x+y dx dy

[/itex]
= 0.125

 

it should be determine Pr (X+Y) < 1/2

 

Yes I mean P(X+Y<1/2)?
The unit square would be

(0,0) (0,1/4) (1/4,0) and (1/4,1/4)

would the limits than be 0< x <= y < 1/4

And the integarl

[itex]
\int_{0}^{\frac{1}{4}} \int_{y}^{\frac{1}{4}} (x+y) dxdy
[/itex]

 

Isn't the unit square just the square with side length 1?

ie

(0,0) (0,1) (1,0) (1,1)

So we are looking at the area where x+y < 1/2

Wouldn't that be in the region (0,0) (0,1/4) (1/4,0) (1/4,1/4) of the unit square ?

so the lmits are

0 < x < 1/4 and 0<y<1/4

[itex]
\int_{0}^{\frac{1}{4}}\int_{0}^{\frac{1}{4}} x+y\text{ } dx dy
[/itex]

 

Is this the area bounded by the triangle with vertices (0,0) (0,1/2) (1/2,0) ?

with limits

0 < x < 1/2 0 < y < 1/2-x

[itex]
\int_{0}^{1/2}\int_{0}^{\frac{1}{2}-x} x+y\text{ }dydx
[/itex]

 

Sorry is the limit of y 1/2-x < y < 1/2

making the integral
[itex]
\int_{0}^{\frac{1}{2}}\int_{\frac{1}{2}-x}^{\frac{1}{2}}x+y\text{ }dydx
[/itex]

 

Thanks alot with your help guys