The Cumulative Effect, Of Adding Many Random Numbers

The question must have crossed many people’s minds, of what the cumulative effect is, if they take the same calculated risk many times, i.e., if they add a series of numbers, each of which is random, and for the sake of argument, if each numbers has the same standard deviation.

The formal answer to that question is explained in This WiKiPedia Article. What the article states, is that ‘If two independently-random numbers are added, their expected values are added, as well as their variance, to give the expected value and the variance of the sum.’

But, what I already know, is that standard deviation is actually the square root of variance. Conversely, variance is already standard deviation squared. Therefore, the problem could be such, that the standard deviation of the individual numbers is known in advance, but that (n) random numbers are to be added. And then, because it is the square root of variance, the standard deviation of the sum will increase, as the square root of (n), times whatever the standard deviation of any one number in the series was.

This realization should be important to any people, who have a gambling problem, because people may have a tendency to think, that if they had ‘bad luck’ at a gambling table, ‘future good luck’ will come, to cancel out the bad luck they’ve already experienced. This is generally untrue, because as (n) increases, the square root of (n) will also just take the sum – of individual bets if the reader wishes – further and further away, from the expected value, because the square root of (n) will still increase. On average!

But, if we are to consider the case of gambling, then we must also take into account the expected value, which is just the average return of one bet. In the real-world case of gambling, this value is biased against the player, and earns the gambling establishment its profit. Well, according to what I wrote above, this will continue to increase linearly.

Now, the question which may come to mind next would be, what effect such a summation of data has on averages. And the answer lies in the fact that the square root of (n), is a half-power of (n). A full power of (n) would grow linearly with (n), while the zero-power of (n), would just stay constant.

And so the effect of summing many random numbers will first of all be, that the maximum and the minimum result theoretically possible, will be (n) times as far apart as they were for any one random number. This reflects the possibility, that ‘if (n) dice were rolled’, they could theoretically all come up as the maximum value possible, or all come up as the minimum value possible. And what this does to the graph of the distribution, is it initially makes the domain of the distribution curve linearly wider, along the x-axis, as a function of (n) – as the first power of (n).

(Updated 05/16/2018 … )

Continue reading The Cumulative Effect, Of Adding Many Random Numbers