Joint Probability FunctionStatistical Independence

[Saladsamurai]

Homework Statement

Homework Equations

If X and Y are statistically independent, then f(x,y) = g(x)h(y) where
g(x) = \int f(x,y) dy
h(y) = \int f(x,y) dx

The Attempt at a Solution

(a)
g(x) = \int f(x,y) dy = \int_{y=0}^{1-x} 6x\, dy
\Rightarrow g(x)=6x(1-x)

and

h(y) = \int f(x,y) dx = \int_{x=0}^{1} 6x \,dx
\Rightarrow h(y)=3

Thus h(y)g(x) \ne f(x,y) and thus X and Y are NOT statistically independent.

Now before I move onto (b) look at the solution that the text gives.

I have no idea what is going on in the upper bound for the h(y) integral? They also went a different route with the solution, but I think that my way should work since it is a definition of independence. But clearly our h(y) functions should be the same. What am I missing?

Thanks,
Casey

 

[vela]

If you sketch the region over which f(x,y)=6x, you'll see it's a triangle with vertices at (0,0), (0,1), and (1,0). From the sketch, you can see that for a given value of y, f(x,y)=6x only for values of x between 0 and 1-y; therefore, the upper limit of the integral should be 1-y, not 1.

 

[Saladsamurai]

Excellent! I should have sketched it. Thanks Vela.

 

[Mark44]

Excellent! I should have sketched it.

I can't think of any situations where this isn't good to do.