Many problems in science, economics, finance, and so on require us to compute conditional expectations. If
and
are
random variables with nice density, then the conditional expectation
can be defined as the orthogonal projection of
on the linear subspace space of all functions of
in Hilbert space of all
(square-integrable) random variables. Thus the conditional expectation can be though of as a function of
with the minimum
distance from
. This justifies the well-known Monte Carlo method of approximating
by generating a sample of (
,
) pairs and regressing
on
by solving a least-squares problem for a polynomial of some chosen degree. In this Demonstration we start with two independent random variables
and
with uniform distribution on [0,1]. As
we take a random variable
for some positive integers
,
, and
. In this case
Mathematica is able to compute the exact formula for
, so that we can assess the accuracy of the estimate by computing the
norm using numerical integration.
K. L. Judd,
Numerical Methods in Economics, Cambridge, MA: The MIT Press, 1998.
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