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Homework Help: Probabilities: The mean of the square

  1. Jan 13, 2009 #1
    1. The problem statement, all variables and given/known data
    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. [itex]<r>[/itex] 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 Pi 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.

    Best regards,
  2. jcsd
  3. Jan 13, 2009 #2
    You gotta 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.

    [tex]\langle r^2 \rangle=\overbrace{\sum ... \sum}^{\rm{n times}}(r_1+r_2+...+r_n)^2 \times \rm{Probability}[/tex]

    Now, how would you simplify that?
  4. Jan 14, 2009 #3
    Thanks, I see it now.
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