Karma

As with any analysis, the basic assumptions must be stated upfront.  So we're going to start with the basic premises that

1. If you smile at someone, they will be inclined to smile at someone else (not necessarily you).
2. Your "soul" will be reborn into another body after you die.  

The variables here are:

1. The distribution of people's ages.
2. The distribution of interactions between people.
3. The distribution of frequencies of smiles.

As an aside, there can always be more variables added, such as if you smile at someone, they may only be 80% inclined to smile at someone else instead of 100%.  But with these variables, any mathematician, statistician, economist, etc. worth his/her salt can come up with simple equations proving karma exists.  In fact, the output of the equations will be how long, on average, it takes for karma to come back to you. I.E. if you smile at someone today, how many years, or how many lifetimes, on average it will take for you to receive a smile, because of that initial smile you gave someone.  I will not derive the equations (as an initial guess I'm pretty sure they are simply partial differentiable equations), because this is just a thought experiment.  The question isn't if you will receive karma - it is when.  It's just math.  However, if you take into account that frowns make someone less inclined to smile, then the question actually does become if, instead of when.

Furthermore, a more complicated (and realistic) analysis of karma could be performed by assuming that smiling at someone else makes a positive impression in their brain.  Perhaps that positive impression will make them more inclined to smile at someone else, or perhaps 15 positive impressions will make them more inclined to perform a "bigger" good deed, such as helping an old lady cross the street.  Then, you wouldn't see a direct effect of your smile in the future (i.e. you wouldn't necessarily receive an equal smile at some point in the future), but perhaps a large good deed will be performed for you.  Again, the equations become more complicated, but any semi-decent mathematician can come up with them.  The idea is that you don't know what kind of results will come from your good deeds, but there will be some.  

Actually, an interesting output of these equations is what is the probability that you will "receive your karma" in this lifetime.  So for example, if you don't believe in assumption #2, then you could see what is the benefit of doing good deeds in this lifetime.  My instinctual guess is that what good karma you receive in this lifetime will only be realized if a large number of people increase the frequencies of good deeds.  This may be a depressing thought, but with the number of people in the world, if only you increase the number of good deeds you do, then there might not be a significant increase in the karma you personally will receive.  A more optimistic idea, however, is that if you take into account the fact that the people you have the most interactions with are probably going to be the ones who are the subject of your good deeds, then the increasing the number of good deeds you do to them will, in fact, probably have a significant impact on your life in the near future.

As with any probabilistic analysis, there will be variance.  That means that if you smile at someone today, you may receive a smile tomorrow, or it may take 1000 years for you to have received that smile.  It is not a guarantee that it will happen at a specific date, or within a certain number of years - there is only a confidence value that it will happen within a certain time range.  For example, if you want to know if a smile will be returned to you within the next 10 years, all you can say is "yes" or "no" with a 95% confidence interval (well, not exactly mathematically, but you get the idea).