Teaser or fun calc equation to surprise husband (physicist/EE) at work

I am a geneticist and unfortunately have not worked much with advanced calc since undergrad. In genetics, as you likely know, a male is denoted as XY and a female as XX. I plan to leave a riddle for my husband and his colleagues at his workplace in order for them to solve. The solution will be an unexpected announcement of the baby's sex.

I am looking for 2 high-level calculus equations - one of which solves to X and the other to Y (and/or two proofs, one of which is true and equals X, and another which is false and equals Y). If anyone would be willing to provide some ideas for this endeavor, the help would be greatly appreciated. Any variations on XX,XY,b(boy),g(girl), etc. would also work. It's a lab full of PhDs, theoretical physicists, and engineers -- so I would imagine the harder the better ;) Bonus points if you can stump them.

Some fields covered by the lab are optics, microwave photonics, radio-over-fiber, quantum computing, etc. Keep in mind they are good, but they are not mathematicians (there may be 1 or 2, but who knows if they'll be around).

-
What about one problem whose solution is $xy$? –  user7530 Sep 11 '13 at 19:10
@user7530 : One problem with solution xy/xx will also work. I will not know which I will be using until next week, but I'd like everything to be ready to go. Thanks for considering my ridiculous request :) Let me know if you ever need any genetics insight. –  Hopkins_Genetics Sep 11 '13 at 19:19
Since $X$ and $Y$ are "automatically used" notation in mathematical statistics, how are they doing related to that field? –  Alecos Papadopoulos Sep 11 '13 at 23:06

I don't know if this will be to your liking, but I admit I had fun constructing it.

First, some word-play in order to explain my motivation: since, as they say, you expect, it is fitting that your husband should calculate an expected value. Also, since whatever the sex of the baby, XX or XY, the first X is given, it is fitting that he should calculate an expected value given X. In other words, a conditional expected value. Then the riddle is: \begin{align} &\text {Your wife expects} \\ &\text {but don't expect}\\ &\text {to find out what}\\ &\text {without some math.}\\ &\text {The first is X}\\ &\text {but what comes next?}\\ &\text {-you need the math}\\ &\text {to find the path.}\\ &\text {Let's leave the prose}\\ &\text {and go verbose}\\ &\text {with symbols and notations:} \end{align}

Let $X$ and $Y$ be two non-negative absolutely continuous, not-independent random variables each ranging in $[0,\infty)$. Calculate $E\Big (E(XY\mid X)\Big)$ given that their joint probability density function is proper and it has the functional form

$$f_{XY}(x,y) = \frac {1}{\sqrt {2\pi}}\frac{2^5}{\pi^2}e^{\frac12 (\ln x-x)}y^2e^{-\frac {4}{\pi}y^2}$$

\begin{align} &\text {This f is strange}\\ &\text {This f seems weird}\\ &\text {but mind can clear the eye,}\\ &\text {the hidden things}\\ &\text {like squares that hiss}\\ &\text {and demons that you know.}\\ &\text {Uncover them!}\\ &\text {-to clear the way}\\ &\text {to what you long to find.}\\ &\text {And when you're there, just don't forget}\\ &\text {to doubly love what you expect.}\\ &\text {But if you still won't understand}\\ &\text {what more to say than }\\ &\text {count the length!} \end{align}

This version leads to a numerical answer that connects to the word "female".

In order to obtain a numerical answer that links to the word "male" you change "to doubly love what you expect" into "to add the one that's coming".

In the second part of the lyrics, there are clues that help focusing the solution approach, and other clues that are critical in order to arrive at the correct number.

Note that the PhD's must know that they search for the sex of the baby.

-
So much more than I could have ever hoped for! Thank you so much, (what I'm assuming is) Dr. @AlecosPapadopoulos! I really appreciate your time and effort, and I'll be sure to let you know the time to an answer! –  Hopkins_Genetics Sep 13 '13 at 17:51
Feels really good that you liked it. If it will indeed be the one that you will use, I certainly want to know whether it was easy or hard for him/them, and whether they found it amusing or not, and indeed, the actual way that they solved it. And I suggest to tell them not to cheat (there are software programs that nowadays solve abstract equations): pencil and paper for this one. Glad tidings. –  Alecos Papadopoulos Sep 13 '13 at 18:05
I'll be sure not to let them break out matlab, etc. Also, to make sure they know the purpose, the riddle will be attached to this lovely book and two balloons (blue and pink): amazon.com/Introductory-Calculus-Infants-Omi-Inouye/dp/… –  Hopkins_Genetics Sep 13 '13 at 18:09