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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).

mrsoltys
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  • Could you please be more specific about how a function $C(x)$ "defines" a random variable? – whuber May 09 '13 at 03:26
  • 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. – whuber May 09 '13 at 19:36
  • 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. – whuber May 09 '13 at 19:51
  • @whuber, I added a non-fluids/environmental example to the description that might alleviate some of the confusion. – mrsoltys May 09 '13 at 20:49
  • Your new question is asked and answered at http://stats.stackexchange.com/questions/14483/intuitive-explanation-for-density-of-transformed-variable/14490#14490. Is this what you were looking for? – whuber May 09 '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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mrsoltys
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