Couples can have any number of sons, and every couple has exactly one daughter. Still, the accepted mathematical solution is an equal gender ratio for the couples' children.
In contrast, "0 sons" is going to describe a full half of all marriages.
So number of expected daughters = 1, number of expected sons = 1. In practice since women can't have an infinite number of children, then this wouldn't be an infinite series, so the real number of expected boys would be lower than one, but there you go…
Now, for the bus case, you get +1 if your bus turns up first, and -1 for every other bus that turns up first. Assume that it is completely random, then: expected + score is: 1/2 1 expected - score is: 1/2 * -1 + 1/4 * -2 + …
Expected + is 0.5, expected - is -1.
> 1 when your bus turns up, -1 for every bus going the other way.
>E[X]/E[Y] = E[X/Y] is not a valid identity.
is completely irrelevant here because it is being used to point out that a non-answer is wrong.
BTW, on a slightly unrelated point, if there's no timetable, but the interval between buses is maintained reliably, the expected waiting time is uniformly distributed over that interval.
If you have to get a second bus, you need to convolve two of those two uniform distributions to find out the distribution of overall journey times. This is a trapezoidal distribution, which is just about analytically manageable.
But a journey with two transfers (3 buses in total) results in a likely overall time distributed according to a uniform distribution convolved with a trapezoidal distribution, which is a very weird non-smooth shape. You can see why people choose to model distributions with Gaussians, which are well-behaved (convolve two Gaussians, get another Gaussian). The Gaussian just lends itself ideally to recursive applications, hence recursive filtering (e.g. Kalman filters).
I suspect this analysis can be carried out and yield quite good results in the gaussian case (a careful analysis might even yield error bounds on the result).
It's a bit less clear that gaussians should be used when e.g. fitting a coordinate to an astronomical feature, which might not actually be symmetrical.
The other useful property that the gaussian has is its separability, in the 2D case. That is unique to the gaussian and counts for a lot.
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[0] https://www.dartmouth.edu/~chance/teaching_aids/books_articl...
It's also possible for the result to be biased because of scheduling. If inbound buses pass every 10 minutes at 16.00, 16.10, 16.20,... and outbound buses at 16.01, 16.11, 16.21, ... you'll usually see an inbound bus first. Though I expect this was not the case here.