Talk:Marchenko–Pastur distribution

The theorem as stated is incorrect. This can easily be verified numerically.

At least for the case M<N, what needs to be fixed is that the \lambda_min and \lambda_\max range need to be scaled by \sqrt N. I don't know if that works in the M>N case.

I'd try to fix this myself by I've never modified wikipedia before, don't have the latex source and don't know how to fix it in the $M>N$ case.

The theorem as stated isn't for a specific M and N. But as M and N approach infinity, the distribution emerges with the specified $\lambda _{\pm }$. As for editing math in wikipedia, you can view Help:Displaying_a_formula. Also when you click "edit", you will see the math markup of the page. 108.35.42.170 (talk) 17:28, 2 November 2013 (UTC)

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Numerics revealed there should be no lambda in the denominator. I removed it. Alex Dubbs — Preceding unsigned comment added by 128.30.51.34 (talk) 16:26, 5 February 2014 (UTC)

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Something isn't right in the cumulative distribution function. We can see if that if we integrate over the probability density function using numerical software. I don't know where the error is as it's hard to find the cdf in textbooks and papers, only the pdf. The formula as stated currently in the article gives probabilities well larger than 1. Winkler (talk) 01:11, 2 April 2021 (UTC)

I believe the correct version is as follows:

${\frac {\sqrt {(\lambda _{+}-x)(x-\lambda _{-})}}{x}}={\frac {\lambda _{-}+\lambda _{+}}{\sqrt {(\lambda _{+}-x)(x-\lambda _{-})}}}-{\frac {x}{\sqrt {(\lambda _{+}-x)(x-\lambda _{-})}}}-{\frac {\lambda _{+}\lambda _{-}}{x{\sqrt {(\lambda _{+}-x)(x-\lambda _{-})}}}}$

For the first integral, see 2.261 of Gradshteyn and Ryzhik; for the second see 2.264-2; for the last see 2.266. The first integral is

$\int {\frac {dx}{\sqrt {(\lambda _{+}-x)(x-\lambda _{-})}}}=-\arcsin {\frac {\lambda _{+}+\lambda _{-}-2x}{|\lambda _{+}-\lambda _{-}|}}$

The second integral is

$\int {\frac {x}{\sqrt {(\lambda _{+}-x)(x-\lambda _{-})}}}\,dx=-{\sqrt {(\lambda _{+}-x)(x-\lambda _{-})}}+{\frac {\lambda _{+}+\lambda _{-}}{2}}\int {\frac {1}{\sqrt {(\lambda _{+}-x)(x-\lambda _{-})}}}\,dx$

which can be calculated further using the first integral. The third integral is

$\int {\frac {dx}{x{\sqrt {(\lambda _{+}-x)(x-\lambda _{-})}}}}={\frac {1}{\sqrt {\lambda _{+}\lambda _{-}}}}\arcsin {\frac {-2\lambda _{+}\lambda _{-}+(\lambda _{+}+\lambda _{-})x}{x|\lambda _{+}-\lambda _{-}|}}$

so putting in the correct values of $\lambda _{\pm }$ gives

$\int {\frac {\sqrt {(\lambda _{+}-x)(x-\lambda _{-})}}{x}}\,dx=-(1+\lambda )\sigma ^{2}\arcsin {\frac {(1+\lambda )\sigma ^{2}-x}{2{\sqrt {\lambda }}\sigma ^{2}}}+{\sqrt {(\lambda _{+}-x)(x-\lambda _{-})}}-|1-\lambda |\sigma ^{2}\arcsin {\frac {-(1-\lambda )^{2}\sigma ^{2}+(1+\lambda )x}{2x{\sqrt {\lambda }}}}.$

Probably this can be simplified. When evaluated at $\lambda _{\pm }$ the antiderivative gives you $\pm \pi \lambda \sigma ^{2}$ if $\lambda <1$ and $\pm \pi \sigma ^{2}$ if $\lambda >1.$ So this integrates out correctly. Of course, this counts as Wikipedia:No original research so should not be included, since I don't know of anywhere in the literature where it's written down. I'll take out the incorrect version. Gumshoe2 (talk) 02:24, 22 February 2024 (UTC)

I was just trying to verify numerically thought I didn't understand something. I will see if I can check published sources. Any suggestions? I am new to this result though. Billlion (talk) 11:57, 14 January 2023 (UTC)