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

[Pacopag]

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?

 

[genneth]

What is $$p(\theta)$$? Is it given?

 

[Pacopag]

Oh ya. Sorry. It is
$$p(\theta) ={2 \over \pi}cos^2({\theta \over 2})$$