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Re: Alternative to bivariate Taylor series

In article <bpd2a7$ce5$1 at>, koshi man <man_koshi at> 

> Is there any expansion to log(1+exp(x-y)) other than
> Taylor series in the integer powers of x and y ? 

There are many other possible expansions. One possibility is that 
Log[1+Exp[x]] is well approximated by truncating the following expansion:

  x UnitStep[x] - Sum[(-1)^i Exp[-i Abs[x]]/i, {i, 1, Infinity}]

Another is to use FindFit to find optimal values for the parameters a, 
b, c, d in the following ansatz:

 x UnitStep[x]+ a Exp[-b Abs[x]] + c Abs[x] Exp[-d x^2] 

> I'm trying to find expectation of above function given
> x, y are random variables with gaussian distributions
> of known  mean and variances.

For numerical parameters, you can use NIntegrate to compute these 
expectation values directly via a straightforward double integral.

Assuming x and y are independent, for the Normal distribution, you can 
reduce the computation of expectation values from a 2D integration to a 
1D integration (which I do not think reduces to a closed form for 
log(1+exp(x-y))). See the Notebook at

> What about other bivariate expansions in Mathematica?

You can always try NIntegrate. Because your function depends linearly on 
a x + b y, for some distributions you will again be able to reduce the 
computation to a 1D numerical integration.

Also, have a look at "Mathematical Statistics with Mathematica" by Colin 
Rose and Murray D. Smith:


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