Regression problem without complete data set I have a medium sized database of hands played at an online poker website. In poker, you starting hand can be classified into one of 169 different hole card combinations (e.g.  AA , 87s which means the eight and seven have the same suit, 87o which means they have different suits). I do not have access to all the sample hands and how much money was made in each hand, but for each hole card combination I have summary stats like the following:

combination          #of hands played         avg profit (in "big blinds"/100 hands)        std dev  (bb/100 hands)
 87s                      245                           23.25                                           54

Ideally I would use the avg profit for each combination as an unbiased estimator of what my profitability for that hand is. However, as you can see the variance is quite large. Also for some combinations the sample size is small because they are rarely played (i.e. they are "folded" pre-flop and not present in the data). However, it should be possible to use data for similar hands to "smooth" the estimate of profitability for each hand.
I want to learn a function that predicts the expected profitability of each of these hands.   I would like to do this by regressing from features I can define on the hole card combination (e.g. sum of ranks, suitedness, connectedness) to the profitability. However, since I don't have the individual samples and only the summary stats above, could I do the following for each combination: 


*

*(for the 87s example above) generate 245 samples from a gaussian distribution  with mean 23.25 and std dev 54.  

*Do this for every hole card combination,  combine the samples and run a linear regression on them.


Does this make sense? Or is there something more principled I can do? 
 A: I really question the Gaussian model for this. Since you have both means and SDs, I'd want to see a plot of the SDs versus the means and see if there is a relationship. (I suspect that higher means have higher SDs.) from the plot, you might then get an idea of whether it would be better to model, say, the log (if the SDs are more or less proportional to the means); or the square roots (if the SDs are proportional to the square roots of the means); or something else. If you can get the SDs more or less stabilized, then you can use the sample sizes as weights in a weighted regression of the appropriately transformed means.
There is a little bit of a bias issue here (e.g., the log of the mean is somewhat larger than the mean of the logs), but the bias would be in the same direction, at least, for all the data so most of it would affect the intercept more than the slopes.
A: If the profit across all the '87s' hands truly follows a gaussian distribution (or at least a distribution that is symmetrical about the mean), then I'm not sure that sampling a distribution 245 times would be necessary. At least not for the purpose of training a prediction model. If your software accommodates weighted samples, you could use a single '87s' entry with a corresponding profit of 23.25. Just give this entry a weight of 245 (or a proportional weight) during the regression.
If your software doesn't accommodate weights, you could duplicate the '87s' sample 245 times. But if we're creating 245 samples anyway, we might as well use your gaussian sampling idea - repeating the gaussian sampling & regression process many times could give you a feel for random fluctuations in the model parameters.
Cheers :)
