Kelly Criterion for game with payoff equal to normal distribution I'm struggling to come up with an answer to a problem I've asked myself, if the answer is out there I must not be searching for the correct things. The problem formulation is this:
You're given the opportunity to play a game, as many times as you want, which has payoff equal to $K$ * $N$($\mu$,$\sigma^2$), where $K$ is a constant of your choice and $N$ an outcome of form a normal distribution with the given parameters that are fixed. You have $X$ available funds and if you reach a negative balance you can't play anymore. What is the optimal choice of $K$, in terms of $X$, $\mu$, and $\sigma$, in a Kelly Criterion sense, meaning that the strategy leads to higher wealth compared to any other strategy in the long run.
 A: Some references.
To supplement the comments of @Dave Harris, here are a cluster of references you might consider using to start further formalising what you've done already. Depending on your mathematical disposition:
1. Kelly's original paper, A New Interpetation of Information Rate (1956). This work stands out in terms of the elegance of the idea and simplicity of presentation.
2. The work of Thorp and associates. See for example:
Thorp, E. O. (2008). The Kelly criterion in blackjack sports betting, and the stock market. Handbook of Asset and Liability Management, 385–428. doi:10.1016/b978-044453248-0.50015-0
3. The work of probabilists in the 60s/70s. Here are some works to illustrate the kind of approach you have outlined using your formalism:
Breiman, L. (1961). Optimal gambling systems for favorable games. Fourth Berkeley Symposium on probability and statistics, I, 65-78
Dubins, L., Savage, L. (1965). How to Gamble if You Must. New York, McGraw-Hill.
Most of these works lie on the academic borders of mathematical statistics/probability and the mathematics of gambling. I have omitted works that have not been vetted by the academic community, such as practical gambling books, because these tend to be written at the level of a numerically literate, but not necessarily mathematically literate audience.  These books also tend to gloss over or use mathematical or statistical concepts somewhat sloppily.
An excellent mathematics of gambling book that does not overly simplify the mechanics of various casino games, and yet remains mathematically rigorous without using more sophisticated machinery (e.g. measure theory etc.) is the following:
Ethier, S. (2010). The Doctrine of Chances: Probabilistic Aspects of Gambling

A general framework for optimal gambling in favourable games.
I have no experience with studying the mathematics of poker, but have spent time reading the literature on using the Kelly criterion with HI-LO card counting in Blackjack.
Here is an extract of a general framework from Thorp (2008) used to illustrate the mechanics of the Kelly criterion asymptotically, and also in studying gambling in mathematics (with its roots in the references cited in 3.).

Imagine that we are faced with an infinitely wealthy opponent who will wager even money bets made on repeated independent trials of a biased coin. Further suppose that on each trial our win probability is $p > 1/2$ and the probability of losing is $q = (1 - p)$. Our initial capital is $X_0$. Suppose we choose the goal of maximising the expected value $\mathbb{E}[X_n]$ after $n$ trials. How much should we bet $B_k$ on each trial? Letting $T_k = 1$ if the $k$th trial is a win and $T_k = -1$ if it is a loss, then $X_k = X_{k-1} + T_k B_k$ for $k = 1, 2, \dots,$ and $X_n = X_0 + \sum^n_{k=1} T_k B_k$. Then
$$\mathbb{E}[X_n] = X_0 + \sum^n_{k=1} \mathbb{E}[B_k T_k] = X_0 + \sum^n_{k=1} (p - q) \mathbb{E}[B_k]$$

Apart from the innovative metaphorical transfer of ideas in communications engineering to gambling, the optimal gambling literature frames the motivation for using the Kelly criterion for betting as a resolution (in an asymptotically optimal sense) to the following trade-off:

*

*That is, maximising expected gain $\mathbb{E}[B_k]$ in each trial by betting the maximum of our resources in each trial, thereby maximising $\mathbb{E}[X_n]$. The probability of bankruptcy (referred to as "ruin" in the literature) in this case is 1 (bankruptcy will occur almost surely).

*The opposing pole would be minimising the probability of eventual bankruptcy, which is equivalent to betting the minimum bet on each trial. This however minimises expected gain and therefore minimises $\mathbb{E}[X_n]$
In this simple framework, it can be shown that the Kelly criterion maximises $\mathbb{E}[\log X_n]$, by betting a fraction $f^* = p - q = 2p - 1$ in each round of one's capital. That is by setting $B_{k-1} = X_{i-1}f^* = X_{i-1}(2p - 1)$.
This is just an introductory framework, and various modifications can be made using this framework for dependent trials with varying win loss probabilities $p_k$ and $q_k$ (e.g. in Blackjack) etc. Furthermore the point @Dave Harris makes about the denomination of capital is important - because in these analyses, capital is assumed to be infinitely divisible, whereas the practical setting violates this assumption, in that we have a fixed minimum currency denomination that cannot be split (e.g. dollars, pounds etc.).
To extend this kind of analysis to poker, which is not played against the House, rather, other players, you might consider finding a way to characterise the expectation/probabilities in each round. This is easier said than done but any informed analysis will almost certainly require some game-theoretic specification of the outcomes given assumptions about other player behaviour. The Ethier book contains a section on Poker, but I cannot personally attest to it, only on the excellent contents of other chapters I have read.
In terms of attempting to estimate parameters $\mu, \sigma^2$ of a Normally distributed cumulative payoff after many rounds in an asymptotic setting (which is a departure from the above class of analysis from the optimal gambling literature, but similar in the use of the Normal distribution), here is a paper which might give you some ideas:
Millman, M. H. (1983). A Statistical Analysis of Casino Blackjack. The American Mathematical Monthly, 90(7), 431.
Lastly, a good informal website in general is:
https://wizardofodds.com/
This website gave fairly accurate computations when I reproduced Basic strategy calculations for Blackjack.
