Joint Probability functions (finding marginals)

[Ryuuzakie]

Homework Statement

$$f\left(x,y\right)=\left\{\begin{array}{cc}\frac{1}{y} ,&\mbox{ if }<br /> 0 \leq x \leq y, 0 \leq y \leq 1 \\ 0,&mbox{ otherwise } \end{array}\right.$$

Find the marginal distributions (pdf and cdf) of X and Y

Find Pr(X+Y>0.5)

Homework Equations

N/A

The Attempt at a Solution

Finding the marginal distribution of X, I get,

$$f_X(x) = \int_{0}^{1} \frac{1}{y} dx$$
$$f_X(x) = ln \left(y\right)\right|_0^1$$

But ln (0) does not exist, and ln (1) = 0... so I'm thinking I'm doing this wrong.

Also, with the second part of the problem, I get:
$$Pr(X+Y \geq 0.5)=1- \left[0.5 ln \left(y\right) - y\right]_0^1$$

And as before, I'm stuck with an ln(0).. any assistance would be appreciated :)!

 

[lanedance]

note that $f_{X,Y}(x,y) [\itex] is only non-zero for the triangle given by x = 0, y= 1 & y=x<br /> <br /> so you may need to rethink your integration limits...$