(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I have a random, uniformly distributed vector with Cartesian components x,y,z. I should calculate the expectation value of the products of the components, e.g. [tex]<x\cdot x>, <x\cdot y>, ..., <z\cdot z>[/tex].

2. Relevant equations

In spherical coordinates the [tex] x,y,z[/tex] components are

[tex]x = \sin(\theta)\cdot\cos(\phi)[/tex]

[tex]y= \sin(\theta)\cdot\sin(\phi)[/tex]

[tex]z = \cos(\theta)[/tex]

[tex]<x\cdot x> =\int f(x)\cdot x^2 dx = const.\times \int x^2 dx[/tex],

because the probability density function is constant of the uniform probability distribution.

3. The attempt at a solution

I think it is useful to convert the Cartesian coordinates to spherical coordinates. So for example

[tex]<x\cdot x> = \int_{0}^{\pi}d\theta\int_{0}^{2\pi}d\phi~~[\sin(\theta)\cdot\cos(\phi)]^{2}[/tex]

I wrote a program for this, and the solution is:

[tex]<x\cdot x> = <y\cdot y> = <z\cdot z> = \frac{1}{3}[/tex]

and the other terms are zero.

But the solution of the above integral is not 1/3.

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# Need help with expectation value !

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