My friend asked me the following:

"If you were to have 2 separate games at different stakes. How would you model them together to combine into one distribution graph. Can you alter the smaller stakes game to be a bigger game with slightly bigger variance and merge graphs while averaging the standard deviation? Or is that just wrong."

I would say that that isn't statistically accurate, but a reasonable estimate. Also it doesn't take into consideration the volume played at each stake, how can you weight it.

I feel like not knowing how to answer this while holding a masters degree in statistics is criminal ;that was just so abstract though, but now I'm genuinely interested in wanting to know how to model this.

What are you trying to determine? What variable do you want a distribution of? If you want winnings, you use BB (in the case of NL poker) instead of dollars and then stakes don't matter. For other games, find the basic betting unit.

Consider that variance (the square of standard deviation) is additive.

And the question is meaningless without specifying the ratio between hands at the various stakes.

So if variance for both levels is V bb/100, and you have stakes where bb=X or bb=Y then the variance for each is XV and YV respectively.

If you played N hands of both stakes, then your total variance would be
NXV + NYV = NV(X+Y)
so your variance per hand would be
V*(X+Y)

To get standard deviation you take the square root so
stdev = sqrt(V*(X+Y)) = sqrt(V) * sqrt(X+Y)
sqrt(V) is your standard deviation in BB for either game, so the standard deviation for equal hands of both stakes is just that times sqrt(X+Y)

Your average win rate/hand for equal numbers of hands would be w*(X+Y)/2

The above is trueish for large samples of hands, due to the central limit theorem. For small numbers of hands, it's not quite true. Like if you played 1 hand of each you can kind of imagine the superposition of the 2 PDFs - you can work out what they would look like if you think about it, but in a general sense you'd have a peak at your win rate for each, with a bell curve centered at each peak. So, your chance of a result in between them would be higher than normal, I think.

This is just my off the cuff thoughts. It feels right to me but I'm not 100% sure.

As NewOldGuy has pointed out above, when you say "pdf", you need to specify (implicitly or explicitly) a random variable for which you wish to know its underlying probability density function.

If you are only interested in knowing certain statistical parameters (rather than the pdf), then RustyBrooks gives you good pointers.

Well, for a "large enough" sample of hands I could make you a PDF from those parameters that would be correct.

For a smaller number I'd have to generate one. And I think you'd probably just add them if you wanted the joint outcome?