# [BioC] RMA biomodality, distribution and transformation

Claus-Dieter Mayer claus at bioss.ac.uk
Fri Jan 25 17:16:16 CET 2008

```Dear Benjamin,

what you are looking for seems to be, what is called the "The Change of
Variables Formula" in English (or the "Dichtetransformationssatz" in
German). Google gave me the following page as one of the first few hits:

http://www.math.uah.edu/stat/dist/Transformations.xhtml

Any text book on the basics of probability theory should cover that too.

Hope this helps

claus

Benjamin Otto wrote:
> Dear bioconductors,
>
> some time ago in one of the discussion about RMA-bimodality Wolfgang
> Huber and Peter Warren pointed out, that a similar distribution could be
> simulated with
>
>
>> n <- 10000
>> z = 20 + exp(c(rnorm(n), 3+rnorm(n)))
>> plot(density(log2(z)))
>>
>
> Now here comes a more mathematical question. Suppose the following
>
>
>> x0 <- rnorm(n)
>> x1 <- x0 + 3
>>
>
> It is quite easy to get the same density in two ways:
>
> for x0 it's super easy:
>
>> plot(density(x0))
>> plot(x0,dnorm(x0))
>>
>
> and for x1 it's still intuitive:
>
>> plot(density(x1))
>> plot(x1,dnorm(x1 - 3))
>>
>
> That's because we are currently only shifting the distribution. But how
> do I transform the x-values for y-value calculation via dnorm() when
> applying the more complex exponential function?
>
>
>> x2 <- exp(x0)
>> plot(density(x2))
>> plot(x2, ..???..)
>>
>
> best regards
>
> Benjamin
>
>
>
>
>
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--
***********************************************************************************
Dr Claus-D. Mayer                    | http://www.bioss.ac.uk
Biomathematics & Statistics Scotland | email: claus at bioss.ac.uk
Rowett Research Institute            | Telephone: +44 (0) 1224 716652
Aberdeen AB21 9SB, Scotland, UK.     | Fax: +44 (0) 1224 715349

```