Probabilities: The mean of the square

[Niles]

Homework Statement

Hi all.

A particle can choose randomly to move in one of these directions:

$$r_1 = (a,0), \quad r_2 = (-a,0), \quad r_3 = (0,a) \quad \text{and}\quad r_4 = (0,-a).$$

These are vectors, not coordinates! I have to find the mean of the square of r, i.e. $<r>$ after n moves, where the particle starts in (0,0).

What I have done is the following:

$$<r^2> = \sum_i {(r_i\cdot r_i)P_i},$$

where P_i is 1/4, because it is random. So I believe the mean of the square of r is a². But my teacher says it is na². I cannot see why he wants to multiply by n, since my method is quite straightforward. Where am I wrong?

Thanks in advance.


Niles.

 

[tim_lou]

You got to have an n somewhere, otherwise, how do you incorporate the fact that it is after n moves? Your expression currently find mean of r^2 after the first step.

For n moves, you'll have n terms and each step has a different possibility. You should think about how to arrive at the corresponding expression.

Hint:
$$\langle r^2 \rangle=\overbrace{\sum ... \sum}^{\rm{n times}}(r_1+r_2+...+r_n)^2 \times \rm{Probability}$$

Now, how would you simplify that?

 

[Niles]

Thanks, I see it now.