Sum of Correlated Exponential RVs

[tpkay]

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

 

[EnumaElish]

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

 

[tacman]

Hi there!

Great question. When dealing with correlated exponential RVs, we can no longer use the convolution method to find the pdf of Y. Instead, we can use the moment-generating function (MGF) of Y to find its distribution.

The MGF of Y is defined as E(e^tY) = E(e^t(X1+X2+X3)). By using the properties of exponential RVs, we can simplify this to E(e^tX1)E(e^tX2)E(e^tX3). Since X1, X2, and X3 are correlated, we can use their joint MGF to find the MGF of Y.

Once we have the MGF of Y, we can use it to find the moments of Y and therefore, the pdf of Y. Alternatively, we can use numerical methods such as Monte Carlo simulation to estimate the pdf of Y.

I hope this helps. Happy studying!