Probability density function of another function

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Discussion Overview

The discussion revolves around the analytical derivation of a probability distribution from a given function defined in radial coordinates. Participants explore the nature of the function, its application in generating a histogram, and the relationship between the function and probability density functions (PDFs).

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant presents a function $$ f(r,\phi)= -\frac{1}{3} -cos(2\phi)(\frac{a^2}{r^2}) $$ defined for an annulus and seeks to derive its distribution analytically.
  • Another participant questions whether the function can be considered a PDF due to its negative values, prompting a clarification that the function itself is not a PDF but is used to generate a histogram of values.
  • A participant suggests that the histogram represents the probability distribution derived from discrete data gathered in Matlab.
  • There is a discussion about the random variables involved, with one participant proposing that a point might be chosen uniformly in the domain of the function to calculate associated values.
  • Clarifications are made regarding the fixed values of the inner and outer radii (a and b) and the uniform distributions of the random variables r and φ.
  • Another participant notes that while the inverse function approach is valid, the presence of multiple pre-images for each value of f complicates the derivation of a closed-form PDF.
  • It is suggested that the cumulative distribution function (CDF) may be more useful than the PDF for this problem.

Areas of Agreement / Disagreement

Participants express uncertainty about the nature of the function as a PDF and the method of deriving a distribution from it. Multiple competing views remain regarding the interpretation of the function and the appropriate approach to derive the distribution analytically.

Contextual Notes

Participants highlight limitations in understanding the distributions of the random variables and the implications of negative values in the function. The discussion does not resolve the mathematical complexities involved in deriving a closed-form PDF.

tx213
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Hi guys, I'm working through a problem right now and would like to pick your brains on some stuff.

I have an function: $$ f(r,\phi)= -\frac{1}{3} -cos(2\phi)(\frac{a^2}{r^2}) \hspace{0.5cm} for \hspace{0.5cm} a<r<b $$. I'm working in radial coordinates so r is the distance from a center and ##\phi## is the angle about that center. Given a particular ## a , b ## which you can think of as the inner and outter rings on an annulus, I can plot this function in Matlab and create a histogram of the values ##f##.

https://www.dropbox.com/s/k1cipzotenl2del/PF.png

My question is whether I can arrive at this distribution analytically?

Thanks in advance for any comments!
 
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Maybe I'm missing something. Is f supposed to be a PDF? It's negative some places so it can't be a PDF. Can you explain what histogram and distribution you are talking about.
 
FactChecker said:
Maybe I'm missing something. Is f supposed to be a PDF? It's negative some places so it can't be a PDF. Can you explain what histogram and distribution you are talking about.

Hi FactChecker,

f is not a PDF. f is my function. I input ##r## (distance from the center of annulus) and ##a## (inner radius of the annulus). ##b## is the outter radius. When ##r## is between ##a,b##, the values in that region of space is described by ##f(r,\phi)##. That is what the map on the left is showing.

I am using Matlab so everything is discrete. That is, I simply gather up all the elements with their associated values from inside that annulus and made a histogram of that list of data.

That histogram is the probability distribution. I want to derive that analytically. It is analogous to say, finding the probability density function of a sine wave.

I know what the general recipe is. Start with your function. Generally the inverse of the function is the CDF. And you can take the derivative to get to the PDF. I'm wondering if that holds true here as well.
 
I'm still trying to figure out what the random variable is. Here are a couple of my guesses that might help you understand why I am confused.

1) Maybe a point is randomly chosen with uniform distribution in the domain of f and it's polar coordinates are used to calculate the associated value of f. That would make the values of f a random variable with a PDF. But then are a and b fixed? If so, what values of a and b gave the "histogram" you show?

2) Values of r, theta, a, and b, are randomly chosen and f is calculated. But I don't know how r, a, and b and chosen or what their distribution is. Are they uniformly distributed over some intervals?
 
FactChecker said:
I'm still trying to figure out what the random variable is. Here are a couple of my guesses that might help you understand why I am confused.

1) Maybe a point is randomly chosen with uniform distribution in the domain of f and it's polar coordinates are used to calculate the associated value of f. That would make the values of f a random variable with a PDF. But then are a and b fixed? If so, what values of a and b gave the "histogram" you show?

2) Values of r, theta, a, and b, are randomly chosen and f is calculated. But I don't know how r, a, and b and chosen or what their distribution is. Are they uniformly distributed over some intervals?

First, allow me to apologize for being unclear. It was not my intent. I realize what you mean.

(1) Yes, a and b are fixed. In this case, a is 10.4 and b is 16 (ignored the labels on the graph, those represent the size of my matrix, not the actual values.)

(2) The random variables are ##r## and ##\phi##. ##r## is uniformly distributed between a and b while ##\phi## is uniformly distributed between ##0## and ##2\pi##
 
Thanks for the clarification. Yes, the same idea of the inverse is correct except that each value of f can have several pre-images whose PDF values should be summed. A formal mathematical treatment would talk about the probabilities of pre-images of intervals in the range of f. I don't know if it is practical to get a closed-form PDF equation for this problem.

On a related issue: Knowing the PDF is not as useful as knowing the CDF since the only positive probabilities are for a range of values of f. The CDF gives those probabilities directly while the PDF requires an integration. I would suggest putting your histogram data into a CDF form if you decide to use that data directly.
 

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