```Date: Feb 15, 2013 8:30 PM
Author: Paul
Subject: Variance of the recursive union of events

I am studying a reliability paper "Analytical propagation ofuncertainties through fault trees" (Hauptmanns 2002).  Unfortunately,I cannot find an online copy to link to.The paper expresses the variance of the union of two events in a waythat doesn't seem to be consistent withhttp://en.wikipedia.org/wiki/Variance#Weighted_sum_of_variables, atleast to my (rather novice) eyes.Using a simplification of the notation in the paper, consider varianceof the recursive relationship:0) c(n) = u(n) + c(n-1) - c(n-1) u(n)for n=1,2,... and c(0)=0.  All c(n) and u(n) values representprobabilities i.e. lie with [0,1].  Furthermore, in the aboveexpression (0), u(n) and c(n-1) are independent.In evaluating the variance of (0), the indices are rather meaningless,as we are completely focused on the right hand side of the equation.I only include them in case a reader has access to the paper.  Thevariance of (0) is presented as:1) var c(n) = var u(n) + var C(n-1) + var[ c(n-1) u(n) ]                         - 2 cov[ u(n) , c(n-1) ]                         - 2 cov[ c(n-1) , c(n-1) u(n) ]According to the above wikipedia page, however, it should be:2) var c(n) = var u(n) + var C(n-1) + var[ c(n-1) u(n) ]                         - 2 cov[ u(n) , c(n-1) ]                         - 2 cov[ u(n) , c(n-1) u(n) ]                         - 2 cov[ c(n-1) , c(n-1) u(n) ]Since u(n) and c(n-1) are independent, their covariance disappears, so(2) becomes:3) var c(n) = var u(n) + var C(n-1) + var[ c(n-1) u(n) ]                         - 2 cov[ u(n) , c(n-1) u(n) ]                         - 2 cov[ c(n-1) , c(n-1) u(n) ]This still differs from (1).  It is plausible that (1) is a typo,though not all that likely.For someone who does this a lot, I imagine that the logic above iselementary.  Thanks for any confirmation on the above.
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