Hi,
I am trying to work out the expected value of a RV z. After some
manipulation, I end up with the following integral to
solve:
\int{zsin{P(z)]^{2}cos(P(z))^{2}}dz
(I think I got that right: integral z*sin2(P(z))*cos2(P(z))dz,
where
sin2 and cos2 mean square of sin and cos, respectively). P(z) is the PDF
of z, and as such is just a Gaussian PDF.
So I am trying to solve this integral, but so far without much success.
My thoughts went as follows:
1.- sin2(P(z))*cos2(P(z)) = (1/4)*sin2(2*P(z))
2.- 2*sin2(2*P(z)) = 1-cos(4*P(z))
So I arrive at the sum of two integrals. One is immediate, and
the other is
\int z
cos(\frac{4}{\sqrt{2\pi}\sigma}exp(\frac{-z^{2]}{2\sigma}})dz,
which I
guessed I could solve by parts (letting z=u and cos(...)=dv). The
problem is I can't integrate something that looks like cos(exp(-z^2)),
and if I reverse the choice of the parts transformation, I don't benefit
at all from the new z^{2] term!
So I thought about a substitution. The only one that made sense to me
was e^{-z^2}=t, but working out the dz/dt bit is problematic (just try
to get the z out of the exponential...).
I guess that my question is... has anyone dealt with this before? I
can't find it in any of the tables at hand, and I suppose that this sort
of thing _must_ have occurred before!!
Thanks
Xose
--
Xose
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