(Variance is “expected squared difference between observation and its prior expected value”, i.e. variance as a concept is closely linked to the mean and not so closely linked to the median or mode. So if you’re talking about “average” and “variance” and the average you’re talking about isn’t the mean, I think at best you’re being very confusing, and possibly you’re doing something mathematically wrong.)
I’m sorry that you are confused. I promise that I really do understand the math.
In repeated addition of random variables, all of these have a close relationship. The sum is approximately normal. The normal distribution has identical mean, median, and mode. Therefore all three are the same.
What makes Kelly tick is that the log of net worth gives you repeated addition. So with high likelihood the log of your net worth is near the mean of an approximately normal distribution, and both median and mode are very close to that. But your net worth is the exponent of the log. That creates an asymmetry that moves the mean away from the median and mode. With high probability, you will do worse than the mean.
The comment about variance is separate. You actually have to work out the distribution of returns after, say 100 trials. And then calculate a variance from that. And it turns out that for any finite n, variance monotonically increases as you increase the proportion that you bet. With the least variance being 0 if you bet nothing, to being dominated by the small chance of winning all of them if you bet everything.
(Variance is “expected squared difference between observation and its prior expected value”, i.e. variance as a concept is closely linked to the mean and not so closely linked to the median or mode. So if you’re talking about “average” and “variance” and the average you’re talking about isn’t the mean, I think at best you’re being very confusing, and possibly you’re doing something mathematically wrong.)
I’m sorry that you are confused. I promise that I really do understand the math.
In repeated addition of random variables, all of these have a close relationship. The sum is approximately normal. The normal distribution has identical mean, median, and mode. Therefore all three are the same.
What makes Kelly tick is that the log of net worth gives you repeated addition. So with high likelihood the log of your net worth is near the mean of an approximately normal distribution, and both median and mode are very close to that. But your net worth is the exponent of the log. That creates an asymmetry that moves the mean away from the median and mode. With high probability, you will do worse than the mean.
The comment about variance is separate. You actually have to work out the distribution of returns after, say 100 trials. And then calculate a variance from that. And it turns out that for any finite n, variance monotonically increases as you increase the proportion that you bet. With the least variance being 0 if you bet nothing, to being dominated by the small chance of winning all of them if you bet everything.