Log-Normal Distribution

From Stanford Wong's BJ21

Posted by MathProf on 16 Dec 1997, at 5:45 a.m., in response to Does it just happen?, posted by ML on 15 Dec 1997, at 7:20 p.m.

Let me try to work on it a little more.

For this discussion, I will begin with a coin tossing problem, with p+q=1 (no pushes.) To keep the opriginal formulas simpler.

You Had the right equation for the Bankroll B(N) that you would have after N plays. As you saw from it, it is bit messy in that it has the key paramters in the exponents. It is simplier to look at its logarithm:

 (1)     log B(N) =    ( 1+f )   * W  + (1-f) * L  

 (2)              =    ( 1 + f ) * W  + (1-f) * (N-W) 

Suppose we want to pick an f whihc will maiximize B(N). Since log is an increasing fct, we may maximize B(N) by maximizing log(B(N), which we do by comuting its derivative and setting it to 0. (ALternatively, you could take the derivative of the original B(N). The easiest way to do that is through "logarithmic differentiation", so you will do the same computation.)
If you take the derivative and set it to 0, you will see the best f occrrs with

  (3)   f   =  (W-L) /  N

One way to think about this. Suppose sevearl different Fractional Bettors are wagering on the same outcomes. If there are 60 Wins and 40 losses, then the 20% bettor would be ahead of all the others at the end. If there were 51 wins and 49 loses, then the 2% bettor would be ahead. If there were 48 wins and 52 losses, then a person wagering "-4%" would do best. If that is not possible, then the most conservative bettor would hav lost less than all the others.

Of course, we dont know ahead of time what W and L will be. But we know their expected value, which is pN and qN. If we put these into equation (3), then we obtain f=(p-q), the ordinary Kelly fraction.

Please note that Log(Bn) is a linear fct of W. Since Expectation is Linear, this means that we can obtain the expecete Logarithm, by plugging in the expeceted value of W. So the Kelly fraction (p-q) also maximizes the "Expected Logarithm".

So I think this is part of the mystery here. B(N) is a compicated fct, but its Log is a linear fct of W (or L). But there is something a little deeper worth pointing out.

Suppose we are interested in the distribution of B(N). Now this will be a complicated to atack directly. However the log[B(N)] is easier. AS noted, it is a Linear fct of W. Now the Central Limit Theorem tells us that Linear Combinations of Normal variable are Normal. So Log(B) is normally distributed! (They say that B has a log-normal distrbituion. Now I would called it an antilog-Normal disribution myself, because it is the antilog of a Normal.)

So Log(B[N]) is Normal with mean equal to the Epxected Logarithm from eqs (1). B = B0 * exp(G) , where G is a random variable which has the Normal distribution.

Approximately half the time, W will exceed its expecetd value and actual Log(B) will be gretaer than than the Expeceted. Approx. half the time, they will be less.

I am not sure that I have really answered your questions. I think the short answer is that with Fraction betting, B has a complex exquation but Log(B) is simple and has desireable properites.