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I would like to define a trigonometric function of the form $f(x)=\alpha \cos^2(P\pi x)$

where I can define $\alpha$ as an amplitude as the function's range, and $P$ as a period related the standard deviation of $x$ values.

Is this possible? I haven't found much help when searching for a link between a cosine function's standard deviation and period. I would be grateful for any help.

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  • $\begingroup$ I don't understand what you are asking. I am also wary of the "standard deviation" terminology. What do you mean by that? $\endgroup$
    – Dylan C. Beck
    Jun 12, 2020 at 14:56
  • $\begingroup$ Well, I guess it could also be "variance". But I am trying to create an empirical model where standard deviations can define the period. I have currently got a trig function by plugging in values and seeing if it fits the data: $cos^2\left(\frac{\pi \sigma _y}{\sigma _x}x\right)$. However, as you might have guessed, it doesn't work for all data sets, and I feel it's 'unmathematical'. So, I would like a way of being able to define the period using standard deviation 'mathematically'. I hope this helps and thanks for your reply btw. $\endgroup$
    – Tom Allen
    Jun 13, 2020 at 0:26
  • $\begingroup$ Oh, so, you really did mean the "standard deviation" in the statistical sense. That sounds quite tough. $\endgroup$
    – Dylan C. Beck
    Jun 13, 2020 at 1:15
  • $\begingroup$ That's exactly it! And yeah, I haven't had much luck so far. $\endgroup$
    – Tom Allen
    Jun 13, 2020 at 1:48

1 Answer 1

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I have found an answer from the academic paper A Cosine Approximation To The Normal Distribution: enter image description here

This can be generalised as...

${\displaystyle \sigma =\sqrt{\int _b^a\left(x-\mu \right)^2f\left(x\right)dx\:}}$

With the standard deviation you would just solve for $P$ in my above equation.

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