I'm reading the conditional distributions section of Probability and Random Processes by Grimmett and Stirzaker and I've come across a brief exercise I can't seem to figure out.
We're given earlier that $f_{X}(x)=\int_{-\infty}^{\infty}f(x,y)dy$ and the example states that X and Y have joint density function
$f_{X,Y}(x,y)=\frac{1}{x}$ for $0 \leq y \leq x \leq 1$
And then it says: show for yourself (exercise) that $f_{X}(x)=1$ if $0 \leq x \leq 1$.
I think I must be being stupid, but I keep getting:
$f_{X}(x)=\int_{-\infty}^{\infty}f(x,y)dy=\int_{0}^{1}\frac{1}{x}dy=\left [ \frac{y}{x} \right ]_{0}^{1}=\frac{1}{x}$. Why is this wrong?
Thanks for your help in advance.
We want to "integrate out" $y$. Note that the joint density is $\frac{1}{x}$ in the part of the unit square below the line $y=x$. So we integrate the constant $\frac{1}{x}$ from $y=0$ to $y=x$. Not from $y=0$ to $y=1$. Above the line $y=x$, we have $f(x,y)=0$.