# How can we plot an equation in 3 variables on a 2D plane in context with functions of pair of Random Variables?

I came across the following question:

The Question

I tried solving it, the following is my attempt:

$$P[W\le w] = P[XY\le w] = P[Y\le w/X]$$

And then I simply double integrated keeping the limits of X on the outer integral and between 0 and 1, and that of Y between 0 and $$w/x$$. But the answer is wrong 'cause finally I ended up calculating $$[w*ln(x)]$$ between 0 and 1.

The solution as given in Probability and Stochastic Processes-Roy D. Yates

1. Why is my answer wrong?
2. What I'm not able to understand is that the plot given should be a 3D plot, how can it be represented in a 2D plane?

(1) $$\mathsf{P}(Y\le w/X)=\int_0^1\int_0^{1\wedge w/x}1\,dy\,dx=w(1-\ln w)$$ for $$w\in [0,1]$$.
(2) It is a level curve of the function $$f(x,y)=xy$$.
• $a\wedge b = \min(a,b)$; What curve are you talking about? – d.k.o. Mar 22 at 17:06