# Level curves and functions of pair of Random Variables?

I came across the following question:

I tried solving it, the following is my 1st attempt:(2nd method at the end)

𝑃[𝑊≤𝑤]=𝑃[𝑋𝑌≤𝑤]=𝑃[𝑌≤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:

Why is my answer wrong? 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, is it the level curve?

Do we first have to plot the level curve and then evaluate the integral over the concerned area?

2nd Attempt:

$$F_W(w)=P[W<=x]=P[X*Y<=w]=P[Y<=w/X]=F_Y(w/X)$$

So now I've to find the CDF of Y and plug in w/X to get the answer.(Is this approach correct?)

$$F_Y(y)=\int_{-\infty}^{\infty} f_{X,Y}(x,y) dx=\int_{0}^{1} dx=1$$

$$F_Y(y)=\int_{-\infty}^{y} 1*dy=\int_{0}^{y} dy=y$$

$$F_Y(w/X)=w/X=F_W(w)$$, but this should be only in terms of w not in X? What to do now?

• I notice you have now posted several questions that are not really about statistics: they are about the basic aspects of double integration. Might I suggest consulting a Calculus textbook? That would likely be a better use of your time than trying to work out the same concepts through the help of various answers to miscellaneous questions. Our format is a great tool for filling in gaps in one's understanding, but it's probably not as effective for introducing people to basic concepts and techniques. – whuber Mar 23 at 13:48