0
$\begingroup$

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?
$\endgroup$
0
$\begingroup$

(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 $f(x,y)=xy$.

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.