Solution to Random Walk Problem | n_i

[arcTomato]

Homework Statement

Prove that the root mean square deviation for a walk involving the sum of different numbers n of steps of length λ

Relevant Equations

random walk

I don't have any idea to prove this
$n_i$is number of trial, right?

 

[haruspex]

$n_i$is number of trial, right?

$n_i$ is the number of steps in the i^th trial

 

[arcTomato]

I got it.
So,I think (4-12)means $σ=Σ_in_iλ_i^2$, but I don't understand what is this

 

[haruspex]

I got it.
So,I think (4-12)means $σ=Σ_in_iλ_i^2$, but I don't understand what is this

You need to start from the other end.
You have a random walk in which n₁ steps are of length λ₁, n₂ steps are of length λ₂, and so on. I presume you are to take these as all being in the same straight line.
At the end of this walk you are at distance X from where you started. Can you write an expression for σ, the root mean square of X?

 

[arcTomato]

Can you write an expression for σ, the root mean square of X?

uhh, I don't know how.
$σ^2=<X^2>=<(n_1λ_1+n_2λ_2+,,,,,n_iλ_i)^2>=<(n_1λ_1)^2>+<(n_2λ_2)^2>+,,,,,,$
like this??

 

[haruspex]

uhh, I don't know how.
$σ^2=<X^2>=<(n_1λ_1+n_2λ_2+,,,,,n_iλ_i)^2>=<(n_1λ_1)^2>+<(n_2λ_2)^2>+,,,,,,$
like this??

No, you need some probabilities in there. Each step can be either way.

 

[arcTomato]

probabilities??like this??