Multivariate probability density - Help with an integral

Question

I'd like some help solving this problem about multivariate probability densities.

Let the random variables X and Y have the joint density f(x,y) = 1/y for 0 < x < y < 1 and 0 otherwise. Find P(X+Y > 0.5).

My problem is that any way I know of to set up the double integral, I get an integral of 1/y (or, switching to variables s and t, 1/t) with a lower bound of integration of 0, but such an integral doesn't converge. If I didn't know that the answer was 0.6534, I would assume that the lower bound of integration in the vertical direction should be x, but that produces a result that includes ln x, not a constant.

I emailed my professor asking for help, but her response was not helpful to me. Maybe it will be to you. She said, "you missed the condition for the density to be positive, so you have to integrate over a different area. And to make things simpler, calculate P(X+Y<0.5) and subtract from 1."

I understand why it's often easier to find a complementary probability and subtract that from 1, but here I don't see how it would help because I don't know how to make the lower bound of integration in the vertical direction anything other than 0 or x. I must be trying to integrate over the wrong region.

The region that I think is correct is the region of the unit square above the line x+y=0.5. That is, a trapezoid with vertices (0, 0.5), (0, 1), (1, 1), (1, 0), and (0.5, 0).

How would you define the region differently, or how would you integrate differently? Thanks so much for your help.

UPDATE: I figured out that I did indeed have the wrong region of integration. When my professor said I missed the condition for the density to be positive, she was implying that I needed to enforce the condition that Y > X. So I needed to add the line y=x to my graph and restrict the integration to the area above it (but still below y=1, obviously).

Thus, the region of integration became the quadrilateral with vertices (0, 1/2), (0, 1), (1, 1), and (1/4, 1/4). The only way I know how to do integration requires dividing this region into two sub-regions. To divide them horizontally, with dx on the outside and dy on the inside, these are the two double integrals (for the left and right sub-regions, respectively):

$ P(X+Y>0.5) = \int_0^.25 \int_{0.5-x}^1 \frac{1}{y}\,dy\,dx + \int_{0.25}^1 \int_x^1 \frac{1}{y}\,dy\,dx $

The left integral equals 0.25, and the right integral equals 0.75 + 0.25(ln 0.25), for a total of 1 + 0.25(ln 0.25) = 0.6534.

Answer 1

The use of multiple integrals requires keeping track of all constraints. A neat approach works through indicators functions, using as many as needed. The probability $\mathbb{P}(X+Y > 0.5)$ can be represented as \begin{align*}\underbrace{\iint_{[0,1]^2}}_{\substack{\text{maximum range}\\\text{for $x$ and $y$}}} \mathbb{I}_{x+y>0.5}\,\mathbb{I}_{0<x<y<1}\,\frac{1}{y}\text{d}x\text{d}y&=\underbrace{\overbrace{\int_0^1\,\int_0^y}^{\substack{\text{outer limits:}\\\text{range of $y$}}}}_{\substack{\text{inner limits:}\\\text{conditional}\\\text{range of $x$}}} \mathbb{I}_{x+y>0.5}\,\frac{1}{y}\text{d}x\text{d}y\\ &=\int_0^1\,\int_0^y \mathbb{I}_{x>0.5-y}\,\frac{1}{y}\text{d}x\text{d}y\\ &=\int_0^1\,\int_{\max(0,0.5-y)}^y \underbrace{\mathbb{I}_{y>0.5-y}\,}_{\substack{\text{as $x<y$}\\\text{and $x>0.5-y$}}}\frac{1}{y}\text{d}x\text{d}y \end{align*} and \begin{align*}\int_0^1\,\int_{\max(0,0.5-y)}^y \mathbb{I}_{y>0.5-y}\,\frac{1}{y}\text{d}x\text{d}y&=\int_0^{0.5}\mathbb{I}_{y>0.25}\,\int_{0.5-y}^y \frac{1}{y}\text{d}x\text{d}y+\overbrace{\int_{0.5}^1\,\int_{0}^y}^{\substack{\text{no constraint on}\\\text{$y$ when $y>0.5$}}} \frac{1}{y}\text{d}x\text{d}y\\ &=\int_{0.25}^{0.5}\,\int_{0.5-y}^y \frac{1}{y}\text{d}x\text{d}y+\int_{0.5}^1\,\int_{0}^y \frac{1}{y}\text{d}x\text{d}y \end{align*} From there you should be able to conclude about the value of this probability, $1-\log(\sqrt{2})$.