Problem with expected value (Random Walk)

[{ imp }]

Homework Statement

Hello, I was reading Feynman's lectures on physics, and I'm having trouble following some deductions in the part about Probability. The random walk is a problem in which someone starts at x = 0 ant then takes a step forward (x = 1) or backward (x = -1) and after N steps de distance traveled is D_N. I'll atach the images regarding the problem because it's kind of large.

I don't really understand how does he get to the part

D_N² = N

How is that?

Homework Equations

They are all on the images.

The Attempt at a Solution

I'm thinking about <D_N-1²> = N - 1, but I don't know why would that be.

I'm sorry if this doesn't go here or if my post is a little hard to read. Thanks in advance.

 

[Ray Vickson]

Homework Statement

Hello, I was reading Feynman's lectures on physics, and I'm having trouble following some deductions in the part about Probability. The random walk is a problem in which someone starts at x = 0 ant then takes a step forward (x = 1) or backward (x = -1) and after N steps de distance traveled is D_N. I'll atach the images regarding the problem because it's kind of large.

I don't really understand how does he get to the part

D_N² = N

How is that?

Homework Equations

They are all on the images.

The Attempt at a Solution

I'm thinking about <D_N-1²> = N - 1, but I don't know why would that be.

I'm sorry if this doesn't go here or if my post is a little hard to read. Thanks in advance.

If you know some elementary probability theory it is easy. We have D = d1 + d2 + ... + dN, where di = distance moved in step i (di = -1 or +1 with probability 1/2 each). The random variables d1, d2, ... are _independent_. There is a basic theorem in probability that says Variance(sum) = sum(Variance), provided that the terms are independent. Thus, Var(D) = N*Var(d1), and Var(d1) = E(d1 - m1)^2, where m1 = E(d1) = mean of d1. Since m1 = 0 we have Var(d1) = E(d1^2) = (1/2)*(-1)^2 + (1/2)*(1^2) = 1. (Here, E = standard probabilistic notation for "expectation" = "mean"; you might prefer to use < > instead; that is, EX = <X>.)

RGV