Derivation of Box-Muller Transform: Exponential Distrib.

  • Context: Graduate 
  • Thread starter Thread starter rabbed
  • Start date Start date
  • Tags Tags
    Transform
Click For Summary

Discussion Overview

The discussion focuses on the derivation of the Box-Muller transform, particularly examining the interpretation of the joint distribution and the role of coefficients in the exponential distribution. The scope includes mathematical reasoning and technical explanation related to probability distributions.

Discussion Character

  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant questions the interpretation of the joint distribution p(x,y) as a product of a uniform distribution and an exponential distribution, noting that the standard form of an exponential distribution includes a coefficient k.
  • Another participant asserts that a constant is missing in the initial interpretation.
  • A third participant provides a derivation starting from the joint distribution in Cartesian coordinates, transitioning to polar coordinates and explaining the appearance of the coefficient through a change of variables.
  • A later reply expresses gratitude for the clarification provided by the third participant.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the interpretation of the coefficient in the exponential distribution, with some asserting its absence and others providing derivations that include it. The discussion remains unresolved regarding the initial question about the coefficient.

Contextual Notes

The discussion highlights potential limitations in understanding the derivation, particularly concerning the definitions and assumptions related to the exponential distribution and the transformation process.

rabbed
Messages
241
Reaction score
3
In derivation of the box-muller transform, the joint distribution p(x,y) = e^(-r^2/2)/(2*pi) is interpreted as the product of a uniform distribution 1/(2*pi) and an exponential distribution e^(-x/2), but isn't an exponential distribution defined as k*e^(-k*x)? What happened to the coefficient?
 
Physics news on Phys.org
You are missing a constant.
 
derivation: Start with [itex]\frac{1}{2\pi}e^{\frac{-x^2-y^2}{2}}dxdy[/itex]. Change to polar coordinates. [itex]\frac{1}{2\pi}e^{\frac{-r^2}{2}}<br /> rdrd\theta[/itex]. For you picture [itex]s=r^2,\ so\ ds=2rdr,\ or\ rdr=\frac{ds}{2}[/itex]. There's the coefficient.
 
Thanks mathman :)
 

Similar threads

Graduate Why is the MGF the Laplace transform?
  • · Replies 6 ·
Replies
6
Views
3K
Graduate How to derive the sampling distribution of some statistics
  • · Replies 3 ·
Replies
3
Views
2K
Graduate The normal equivalent for a discrete random variable
  • · Replies 25 ·
Replies
25
Views
3K
Undergrad Ways to put n balls in m boxes such that exactly k boxes are empty?
  • · Replies 29 ·
Replies
29
Views
6K
MHB How Do You Minimize Mean Square Deviation in Dice Roll Predictions?
  • · Replies 4 ·
Replies
4
Views
2K
Graduate How Is the CDF Expressed for a Uniform Distribution on a Circle?
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad Probability of White Ball in Box of 120 Balls: Solved!
  • · Replies 3 ·
Replies
3
Views
3K
Graduate Deriving the standard normal distribution
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad On pdf of a sum of two r.v.s and differentiating under the integral
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Variable transformation for a multivariate normal distribution
  • · Replies 1 ·
Replies
1
Views
2K