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 $f(x,y)=xy$.

  • $\begingroup$ A few doubts: What's the upper limit of the inner integral? The level curve doesn't matched with the solution figure. Also please answer the doubts marked as 1. and 2. $\endgroup$ – Adarsh Kumar Mar 22 at 16:56
  • $\begingroup$ $a\wedge b = \min(a,b)$; What curve are you talking about? $\endgroup$ – d.k.o. Mar 22 at 17:06
  • $\begingroup$ What does 1^w/x means(The upper limit of inner integral)? I mean this curve:mathinsight.org/level_sets, the one you've attached $\endgroup$ – Adarsh Kumar Mar 22 at 17:10
  • $\begingroup$ 1. See the previous comment. 2. The link gives examples of level curves/sets (you asked how a 3d function can be represented in a 2D plot). $\endgroup$ – d.k.o. Mar 22 at 17:14

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