Bivariate transformation using CDF method

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Discussion Overview

The discussion revolves around the use of the cumulative distribution function (CDF) method to derive the probability density function (PDF) for a transformation involving two independent uniform random variables, U and V. The participants explore the mathematical relationships between these variables and the implications for calculating the PDF of X, defined in terms of U and V.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant presents the transformation equations for X, Y, and Z and seeks to find X_pdf(x) using the CDF method.
  • Another participant suggests expressing the probability P(sqrt(1-V^2)*cos(U) < x) as a condition on V, leading to an integral with respect to U.
  • Several participants discuss the correctness of various transformations and integrals, with some expressing confusion about the implications of uniform distributions and the integration limits.
  • There is a proposal to graph the relationship between V and U, leading to conditions for real solutions based on the values of x.
  • Participants explore the need to integrate over the joint distribution of U and V, with some suggesting that the integration limits should be adjusted based on the conditions derived from the transformations.
  • There is a discussion about the application of the Leibniz integral rule for differentiating integrals, with some participants questioning the validity of certain expressions involving absolute values.
  • One participant attempts to clarify the relationship between the probabilities of V and |V|, leading to a discussion about the symmetry of the uniform distribution.
  • Another participant highlights the importance of adjusting the integration range based on the conditions derived from the transformations, particularly when cos(u) < x.
  • There are ongoing clarifications regarding the integration process, with participants checking their understanding of the steps involved in deriving the PDF from the CDF.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and agreement on the steps needed to derive the PDF. Some participants agree on the need to adjust integration limits and apply the Leibniz integral rule, while others remain uncertain about specific transformations and the implications of uniform distributions.

Contextual Notes

Limitations include unresolved mathematical steps, particularly regarding the integration limits and the treatment of absolute values in the context of uniform distributions. The discussion reflects a range of interpretations and approaches to the problem without reaching a consensus.

Who May Find This Useful

This discussion may be useful for individuals interested in probability theory, particularly those exploring transformations of random variables and the application of the CDF method in deriving PDFs.

  • #31
Yeah, that got a bit nasty. Thanks, I'll dig into how that relates geometrically later.
My main wondering was if there was some general way to deal with source RV limits when they consist of variables, but I guess it's highly specific to the geometry.

I think you might have missed something I wrote above, because I edited it while you wrote your reply maybe:
For a uniformly distributed angle of a circle with radius r, I calculated X_PDF(x) = 1/(pi*sqrt(r^2-x^2)), Y_PDF(y) = 1/(pi*sqrt(r^2-y^2)). I guess that means that I can randomize a bunch of x-coordinates according to X_PDF(x) and for each x, I set y = +/-sqrt(r^2-x^2) and this will give me a uniform distribution of points around the circle?
Would the equivalent for a sphere be to solve X_PDF(x) and Y_PDF(y) by building on the knowledge we've gathered in this post, then randomize an x and y-coordinate according to those distributions and set z = +/-sqrt(R^2-Y^2-X^2)?
Y_PDF(y) = 1/(2*R)
Z_PDF(z) = 1/(2*R)
X^2 = R^2 - Y^2 - Z^2
 
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  • #32
rabbed said:
Yeah, that got a bit nasty. Thanks, I'll dig into how that relates geometrically later.
My main wondering was if there was some general way to deal with source RV limits when they consist of variables, but I guess it's highly specific to the geometry.
It can all be done without the geometry, but it takes a lot of checking ranges to make sure the integrals are right. Where possible, a picture will make things much clearer. Did you try drawing this one?
rabbed said:
For a uniformly distributed angle of a circle with radius r, I calculated X_PDF(x) = 1/(pi*sqrt(r^2-x^2)), Y_PDF(y) = 1/(pi*sqrt(r^2-y^2)).
Not sure how you are defining the distribution. A uniform distribution over the circle will lead to a uniform distribution of angle from the centre, but a non-uniform distribution of radius. But you might mean some other distribution over the circle. Your PDF formula doesn't look right, it doesn't integrate to 1.
 
  • #33
Okay
Yes, I graphed it on www.desmos.com, but I think i'll give it up for now.

Sorry, should have been more clear, I calculated it using:
Source RV: A_pdf(a) = 1/(2*pi) 0 < a < 2*pi
Destination RV: X = R*cos(A)
If I integrate the PDF between -r to r it results in 1.
I'm learning about the inverse CDF method now so I can check if the generated x-coordinates (non-uniform along the x-axis) with associated y-coordinate +/-sqrt(1-x^2) produces points according to a uniform angle-distribution.

I would like to do the same in 3D, randomize according to some distribution(s) along for example the Y and Z-axes, set x=+/-sqrt(R^1-Y^2-Z^2) to produce uniform points on the surface of a sphere of radius R. Does it seem resonable and do you think I can get the distribution(s) using the conditions from my last post? I tried earlier today, but the result didn't seem right so i'll go over it again.
 
  • #34
Or can I just distribute y and z uniformly over [-R, R] and set the associated x coordinate = +/-sqrt(R^2-Y^2-Z^2) to get points that are also uniform over the surface area of the sphere?
 

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