Derive Convolution Expression for Z_PDF(z)

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SUMMARY

The discussion focuses on deriving the convolution expression for the probability density function (PDF) of the sum of two independent random variables, Z = X + Y. The participants explore the relationship between Z_PDF(z), X_PDF(x), and Y_PDF(y), emphasizing that Z_PDF(z) must account for all combinations of X and Y that sum to z. They conclude that the correct formulation involves integrating the joint PDF of X and Y over the appropriate region, leading to the convolution formula: Z_PDF(z) = ∫ X_PDF(x) * Y_PDF(z-x) dx. The necessity of proper integration limits and the independence of X and Y are also highlighted.

PREREQUISITES
  • Understanding of probability density functions (PDFs)
  • Familiarity with convolution in probability theory
  • Knowledge of integration techniques, particularly for joint distributions
  • Basic concepts of independent random variables
NEXT STEPS
  • Study the convolution theorem in probability theory
  • Learn about joint probability distributions and their properties
  • Explore the application of the Leibniz integral rule for differentiating under the integral sign
  • Investigate the derivation of convolution formulas for multiple independent random variables
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Statisticians, data scientists, and mathematicians interested in probability theory, particularly those working with random variable transformations and convolution operations.

  • #61
As far as I can tell, you aren't presenting any logical arguments. You are conjecturing various formulas and asking for criticism of them. That's a permissible approach in the early stages of an investigation, but you should follow-up a conjecture by testing it with some simple examples instead of relying on my comments. it's ok to make conjectures by resorting to "magic" - such as writing down symbols like "dx/dx" without asking what they symbolize. But you should proceed to working specific examples that force you to make specific interpretations. (I'm about to get busy for a few days with the jobs of being executor of an estate, so I'm not going to have time to criticize a hundred different conjectures.) - In fact I just got a phone call and I must leave right now.
 
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  • #62
I think the essence here is that for multiple variables/dimensions we need to use line/surface integrals
and that it starts to make more sense to use the joint PDF of the source RV's.
XY_PDF(x,y)*sqrt(dx^2 + dy^2) = X_PDF(x)*|dx| * Y_PDF(y)*sqrt(1 + (dy/dx)^2)

Hm, maybe regular multidimensional integration comes in when considering inequalities, like Z < X+Y.

I'll keep exploring. Thanks for all help!
 

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