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## Homework Statement

I GOT all values...but I have trouble explaining.....if someone could suggest how, thank you! Please check the red fonts which shows where I got stuck!

Suppose we call x the position of the boy and each step he takes (east or west ONLY) is of 1m. So, for an "n" step random walk he takes, the n+1 possible positions that the boy can end up at are given by:

x = 2k +n, where k = 0, 1 ...n

and the likelihood of ending up at any one of these is determined by the probability: p(k) = 1/(2^n) (n k) where (n k) = n! / [(n-k)!(k!)] and k = 0, 1 ... n.

(a) Explain why the formula p(k) should be true.

(b) Compute <x> after an n-step walk. Why should <x> have this value?

(c) compute [<x^2>]^0.5 after n step walk. Suppose all n steps were in the same direction, what would be [<x^2>]^0.5?

## Homework Equations

x = 2k +n, where k = 0, 1 ...n

p(k) = 1/(2^n) (n k) where (n k) = n! / [(n-k)!(k!)] and C is a normalization constant and k = 0, 1 ... n.

## The Attempt at a Solution

([/B]a) B/c Sigma (k=0 to n) p(k) = 1 for normalization, if n =1, supposing x = -1, k would be equal to 1...giving a p(1) = 1/2 (1/1) = 1/2. This is the explanation I put...but, I don't understand in the case when n=1, x = 1, in which k = 0, giving p(0) = 1/2 (1/0) = ????. Could someone hint why the formula is true?

(b) I obtained the value by:

<k> = Sigma (k=0 to n) k*p(k)

= Sigma k (n k) 1/(2^n)

= 1/(2^n) Sigma k (n k) = 1/(2^n) n * 2^(n-1) = n/2

therefore...<x> = 2<k> - n

= 2(n/2) - n = 0....Why did I get the value of zero, if uncertainty is possible????? Is it b/c there is an equal chance of being east or west of the initial position the boy was in the first place?

(c) Similarly, I derived <k^2> which I got n(n+1)/4. So:

<x^2> = <(2k+n)^2> = <4k^2 - 4kn + n^2> = 4<k^2> - 4n<k> + n^2 into which I input the results I had from the above...and got <x^2> = n

So...[<x^2>]^0.5 = n^0.5. However, what should I do if all n steps were in the same direction? Because that is once in all possible probabilities, so, should it be 1/n^0.5? I am really stuck here!