Sum of Correlated Exponential RVs

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SUMMARY

The discussion focuses on the challenge of determining the probability density function (pdf) of the sum of correlated exponential random variables (RVs), specifically Y = X1 + X2 + X3, where X1, X2, and X3 are exponentially distributed. When these RVs are independent, their pdf can be derived through convolution; however, the approach changes when they are correlated. Suggested methods for addressing this problem include simulation and polynomial fitting to the simulation data. Relevant literature includes Makarov's 1981 work on distribution functions and several additional references that provide further insights.

PREREQUISITES
  • Understanding of exponential random variables (RVs)
  • Knowledge of probability density functions (pdfs)
  • Familiarity with convolution in probability theory
  • Basic skills in statistical simulation techniques
NEXT STEPS
  • Explore simulation techniques for correlated random variables
  • Learn about polynomial fitting methods for statistical data analysis
  • Study the implications of correlation on the distribution of sums of random variables
  • Review Makarov's 1981 paper on distribution functions for deeper insights
USEFUL FOR

Statisticians, data scientists, and researchers in probability theory who are dealing with correlated random variables and their distributions will benefit from this discussion.

tpkay
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Hi All :)

say Y = X1 + X2+ X3, where X1, X2 and X3 are each exponentially distributed RV. This makes Y also a RV. If X1 and X2 and X3 are independent, the pdf of Y can be found by the convolution of the individual pdfs.

What if X1, X2 and X3 are correlated? How do we go about finding the pdf of Y?

many thanks in advance

:-p
 
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1. Simulate,
2. Fit a polynomial to simulation data.

Makarov, G. (1981) "Estimates for the distribution function of a sum of two random variables when the marginal distributions are fixed," Theory of Probability and its Applications, 26, 803-806.

See others under References in:
http://www.math.ethz.ch/%7Estrauman/preprints/pitfalls.pdf

Also see:
http://www.merl.com/publications/TR2006-010/
http://ieeexplore.ieee.org/Xplore/l...29369/01327853.pdf?isnumber=&arnumber=1327853
http://arxiv.org/abs/cond-mat/0601189
 
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