Find Expectation Value for Particle Moving in N Steps of Length L

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Homework Statement


A particle moves in a sequence of steps of length L. The polar angle \theta for each step is taken from the (normalized) probability density p(\theta). The azimuthal angle is uniformly distributed. Suppose the particle makes N steps.
My question is how do I find the expectation value (say <z^2> for example).

Homework Equations


Usually for a probability density p(x) we have
<x^m>=\int x^m p(x) dx.

The Attempt at a Solution


I think that I can get the values for one step. eg.
<z^2>=\int_0^\pi (Lcos(\theta))^2p(\theta)d\theta={L^{2}\over 2}
Note: the density p(\theta) is normalized.
I just don't know how to treat N steps. Do I just multiply the one-step result by N?
 
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What is p(\theta)? Is it given?
 
Oh ya. Sorry. It is
p(\theta) ={2 \over \pi}cos^2({\theta \over 2})
 

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