# Difference between revisions of "Talk:Derivation of the Rayleigh Distribution Equation"

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''"The Rayleigh Distribution makes the following simplifying assumptions to the general bivariate normal distribution:"'' is a bit weird because a distribution does not make assumptions. Making the assumption that shots (in the sense of (h,v) coordinates) follow a restricted (circular) bivariate normal distribution implies that the distance to the true COI follows a Rayleigh distribution. | ''"The Rayleigh Distribution makes the following simplifying assumptions to the general bivariate normal distribution:"'' is a bit weird because a distribution does not make assumptions. Making the assumption that shots (in the sense of (h,v) coordinates) follow a restricted (circular) bivariate normal distribution implies that the distance to the true COI follows a Rayleigh distribution. | ||

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+ | :: You're right, the wording is sloppy.<br />[[User:Herb|Herb]] ([[User talk:Herb|talk]]) 20:13, 24 June 2015 (EDT) |

## Latest revision as of 20:13, 24 June 2015

** rev 13:28, 3 June 2015 Herb **

Ok, name of page needs a bit of fixing.

more verbosity in text

I goofed in accuracy section. "PDF" functions are set up wrong. Should be something like "PDF of X as x"

Been about 45 years since I took calculus in college. I'll have to look up conversion from Cartesian to Polar coordinates, but I know the conversion will swizzle to the right answer...

This is topic that I think bears further investigation...

is it \(\sigma_h = \sigma_v\) or \(\sigma_h^2 = \sigma_v^2\) ??

Of course if they are equal then both equations are true. The problem is in pooling the values if:

\(\sigma_h \approx \sigma_v\)

which equation do we use to pool the values?

- \((\sigma_h + \sigma_v)/2\)
- \(\sqrt{\sigma_h^2 + \sigma_v^2}\)

I think \(\sqrt{\sigma_h^2 + \sigma_v^2}\) should be corrected by factor \(\frac{1}{\sqrt{2}}\) since \((\sigma + \sigma)/2 = \sigma\) but \(\sqrt{\sigma^2 + \sigma^2} = \sigma \sqrt{2}\)

In general it would seem that the ratio \((\sigma_h / \sigma_v)\) could be useful as a guide to stay out of trouble. Obviously the ratio should depend on sample size, *n*. Think this is sort of the idea, but it doesn't right (two limits should converge) ...

\( .33\frac{(\sigma_h + \sigma_v)}{\sqrt{n}} \leq (\sigma_h / \sigma_v) \leq 0.75\frac{(\sigma_h + \sigma_v)}{\sqrt{n}} \)

I also really don't like using a \(\sigma\) as a factor in the equation. If you think about radial values then there is a \(\sigma\) which can be calculated from the \(r_i\) values. The two \(\sigma\)'s aren't equal.

The !@#$%^&* literature really messes up calculation of radial standard deviation. needs \(\frac{1}{\sqrt{2}}\) correction.

4.1 Derivation From the Bivariate Normal distribution

... being more repetitive: *"a simple translation of the Cartesian Coordinate System converts the Bivariate Normal distribution to the Hoyt distribution"* is misleading: The bivariate normal distribution does not become the (univariate) Hoyt distribution. After converting (h,v) coordinates to polar (r,azimuth) coordinates, the r-coordinate (distance to true COI) follows a Hoyt distribution.

5.1 Derivation OF Single Shot PDF From the Bivariate Normal distribution

*"The Rayleigh Distribution makes the following simplifying assumptions to the general bivariate normal distribution:"* is a bit weird because a distribution does not make assumptions. Making the assumption that shots (in the sense of (h,v) coordinates) follow a restricted (circular) bivariate normal distribution implies that the distance to the true COI follows a Rayleigh distribution.