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

Solution 1:

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.