When betting, investing, etc. one might have several possible outcome distributions that they can allocate their "bets" to. For example, bet on black vs. bet on number 8 or invest in stock fund A vs stock fund B.
For the sake of argument, let's model all of our outcomes as a normal distribution around a multiplier on your initial bet. For example, a money market fund may have a mean of 1.02 / year with a standard deviation of 0.001.
Assume that you will bet your entire bankroll repeatedly (N times) on the same gamble. ie: if you double your money, you bet twice as much next time.
Given two possible distributions with different means and standard deviations, how would you select which one to bet your money on N times in order to maximize your final return?
Presumably if one of the distributions had a higher mean and lower standard deviation, then the decision would be simple. The interesting case is when one of the distributions has higher mean and higher standard deviation than the other.
If $X_i$, $i=1..N$ are independent with means $\mu$ and standard deviations $\sigma$, $R_N = X_1 X_2 \ldots X_N$ has mean $\mu^N$ and standard deviation $(\mu^2 + \sigma^2)^N - \mu^{2N}$.
However, means can be very poor indications of what will happen in the long run when dealing with products of random variables. Typically, most of the mean is the contribution of events of very small probability but very large return. A better indication of what happens is given by $\log R_N = \log(X_1) + \ldots + \log(X_N)$, since sums of iid random variables behave quite nicely (law of large numbers, central limit theorem etc). You might look at http://www.maplesoft.com/applications/view.aspx?SID=127116
You don't really want to use normal distributions here though, because they have nonzero probability of being negative.