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Say I have a random variable (such as concentration) that is defined spatially by some function $C(x)$.

I would like to derive a PDF $(f(C))$ from the concentration function $C(x)$ over some domain of x, so that if I pick a random value of $x$, $f(C)$ describes the probability of $C$.

Here's an example (from http://www.cs.ucr.edu/~ciardo/teaching/CS177/section7.1.pdf). The line segment joining two points on a circle has length $Y$ defined by $Y = 2r \sin{ (Θ/2)}$

if $Θ$ is Uniform$(0,2π)$, the pdf of Θ is $f (θ) = 1/2π$.

The expected (mean) value of Y can be defined as: $E[Y] = \int^{2π}_0 Y(θ) f(θ) dθ$

How could i define the probability density function of $Y$, $f(Y)$? (without assuming that f(Y) has a normal or some other distribution).

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  • $\begingroup$ Could you please be more specific about how a function $C(x)$ "defines" a random variable? $\endgroup$ – whuber May 9 '13 at 3:26
  • $\begingroup$ We may have a terminology issue. Although a spatially-distributed concentration can be considered a random variable, it sounds like you think of it as being determinate and you are considering sampling it randomly (such as uniformly and independently). If it's really the former, you need to describe the properties of the stochastic process producing $C$. If it's the latter, please confirm that you have uniform sampling in mind and not some other spatial sampling distribution. If it's neither, please clarify. $\endgroup$ – whuber May 9 '13 at 19:36
  • $\begingroup$ But what do you mean by "probability"? In your example, at any given time $t$ the concentration is what it is: there's no probability about it. Some crucial idea is still missing from your statement of this problem. If you are looking for the frequency distribution of concentrations, that will depend on the exact extent (and shape in higher dimensions than 1) of the domain, the location of the origin, and--of course--the time. $\endgroup$ – whuber May 9 '13 at 19:51
  • $\begingroup$ @whuber, I added a non-fluids/environmental example to the description that might alleviate some of the confusion. $\endgroup$ – mrsoltys May 9 '13 at 20:49
  • $\begingroup$ Your new question is asked and answered at stats.stackexchange.com/questions/14483/…. Is this what you were looking for? $\endgroup$ – whuber May 9 '13 at 21:13
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Thanks for the help. @whuber pointed me to a post discussing changing variables, which lead me to read a bit on wikipedia.

a change in variables is defined as: $f_Y(y) = \left| \frac{d}{dy} (g^{-1}(y)) \right| \cdot f_X(g^{-1}(y))$

For the example above:

$Y=2 r sin(Θ/2)$

$Θ=2 arcsin{\frac{Y}{2r}}$

$\frac{dΘ}{dY}=\frac{2}{r\sqrt{4-\frac{Y^2}{r^2}}}$, $f(θ)=1/2π$

so $f_Y(y)=\frac{1}{r\pi\sqrt{4-\frac{Y^2}{r^2}}}$

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