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 … )

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The Relationship between Voltage and Energy

Energy is proportional to voltage squared. If we make the assumption that a variable voltage is being fed to a constant load-resistor, then with voltage, current would increase, and current would get multiplied by voltage again, to result in energy.

Sound energy is proportional to sound pressure squared. With increasing sound pressure, minute displacement / compression of air results, which causes displacement to rise, and displacement times pressure is again – energy.

The decibel scale is in energy units, not pressure units. Therefore, if a voltage increases by the square root of two, and if that voltage is fed to a constant load, then energy doubles, which is loosely expressed as a 3db relationship. A doubling of voltages would result in a quadrupling of energy units, which is loosely described as a 6db relationship.

Something similar happens to digitally sampled sound. The amplitudes of the samples correspond roughly to the Statistical concept of Standard Deviation, while the Statistical concept of Variance, corresponds to signal-energy. Variance equals Standard Deviation squared…

Dirk

I should add that this applies to small-signal processing, but not to industrial power-transmission. In the latter case, the load resistances are intentionally made to scale with voltages, because the efficiency-gains that stem from voltage-increases, only stem from keeping current-levels under control. Thus, in the latter case, higher amounts of power are transmitted, but without involving higher levels of current. And so here, voltages tend to relate to power units more-or-less linearly.

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The wording ‘Light Values’ can play tricks on people.

What I wrote before, was that between (n) real, 2D photos, 1 light-value can be sampled.

Some people might infer that I meant, always to use the brightness value. But this would actually be wrong. I am assuming that color footage is being used.

And if I wanted to compare pixel-colors, to determine best-fit geometry, I would most want to go by a single hue-value.

If the color being mapped averages to ‘yellow’ – which facial colors do – then hue would be best-defined as ‘the difference between the Red and Green channels’.

But the way this works out negatively, is in the fact that actual photographic film which was used around 1977, differentiated most poorly between between Red and Green, as did any chroma / video signal. And Peter Cushing was being filmed in 1977, so that our reconstruction of him might appear in today’s movies.

So then an alternative might be, ‘Normalize all the pixels to have the same luminance, and then pick whichever primary channel that the source was best-able to resolve into minute details, on a physical level.’

Maybe 1977 photographic projector-emulsions differentiated the Red primary channel best?

Further, given that there are 3 primary colors in most forms of graphics digitization, and that I would remove the overall luminance, it would follow that maybe 2 actual remaining color channels could be used, the variance of each computed separately, and the variances added?

In general, it is Mathematically safer to add Variances, than it would be to add Deviations, where Variance corresponds to Deviation squared, and where Variance therefore also corresponds to Energy, if Deviation corresponded to Potential. It is more generally agreed that Energy and its homologues are conserved quantities.

Dirk