Thread: Using Taylor expansion for calculating Expected Value

1. Using Taylor expansion for calculating Expected Value

Hello,

This question is motivated by a very helpful approach suggested by BGM for solving an expected value estimation problem recently posted on this site.

BGM has suggested using a Taylor expansion for this. Namely, suppose X and Y are random variables, Y is a function of X, say Y = f(X), and the distribution of X is known (say, it is uniform). We first consider the Taylor expansion of the function f around some value a:

y = f(a) + (xa)f’(a) + (f’’(a)/2)(xa)^2 + …

and then use it to write an approximation (around a = E[X]) of the expected value of Y:

E[Y] = E[f(E[X]) + (X – E[X])f’(E[X]) + (f’’(E[X])/2)(X – E[X])^2 + …].

I have two questions regarding this approach:
(1) In what book/paper can I find results justifying this approach (that is proving that under some conditions the approximation converges to E[Y])?
(2) If the relationship between X and Y is implicit (for example, we only know that X and Y satisfy an equation XY +cos(X+Y) = 1), under what conditions the Taylor expansion for y = f(x) converges to the “true” solution of the equation (actually this question is not probability-related but it is important here)? Again, I would highly appreciate any advice on where an answer could be found.

Thank you.

2. Re: Using Taylor expansion for calculating Expected Value

I found one definition/condition in wiki: (I am not familiar with this)

http://en.wikipedia.org/wiki/Taylor_series
If f(x) is equal to its Taylor series everywhere it is called entire.
http://en.wikipedia.org/wiki/Entire_function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane.
And if you want to exchange the order of limits with the expectation,
you will need Dominated Convergence Theorem, Monotone Convergence
Theorem, or Fatou's Lemma.

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