Please teach me how to prove this random walk problem....

[arcTomato]

Homework Statement

prove that the mean position after a given number of steps is the starting position

Relevant Equations

random walk

I think I can prove equal length version problem, But I am confusing in this case,,,

Please help me!

 

[hutchphd]

If you understand the case of 1 path length, consider the case of two allowed path lengths. If you can do that I believe the generalization will be evident.

 

[arcTomato]

Ok, I tried it like this.
Is it right??

 

[hutchphd]

OK, So how do you show the single length case to give zero..

 

[rude man]

You know the mean of equal-distance lengths is zero. Now take the hint given: you can make those equal-length steps as small as you like!

 

[arcTomato]

Thanks your reply.
If I make the length of one step 0, the mean of length is 0.
$k→0,<k>=0$
So,,,,,I don't know what does this mean :<

 

[hutchphd]

Problem Statement: prove that the mean position after a given number of steps is the starting position
Relevant Equations: random walk

I think I can prove equal length version problem

First things first. Make good on your "I think I can"...and show me
Then we will worry about the way to generalize

 

[arcTomato]

First things first. Make good on your "I think I can"...and show me

Is it wrong this photo??I thought this can prove equal length version.
If this is wrong, I have to withdraw my statement. ;;

 

[hutchphd]

View attachment 246458
Is it wrong this photo??I thought this can prove equal length version.
If this is wrong, I have to withdraw my statement. ;;

Your formula for <k> is correct but how do I know it sums to zero?

 

[arcTomato]

$P(n,k)=P(-n,k)$
So, the pair of numerator are canceled and the one having $k=0$ is uncancelled. this shows the sum is zero, right??

 

[hutchphd]

$P(n,k)=P(-n,k)$
So, the pair of numerator are canceled and the one having $k=0$ is uncancelled. this shows the sum is zero, right??

Exactly. You would formally write that as
$ΣkP(n,k)=ΣkP(n,k)-ΣkP(-n,k)$
$=k Σ(P(n,k)-P(-n,k))=0$,, by symmetry
and you would explicitly delimit the sums (I'm too lazy )

I think you see that the multi-length sum can be manipulated simply to produce the same result

 

[rude man]

You don't need any fancy math.
Say you have two steps of different lengths x1 and x2. How about dividing each of those into smaller, equal-length steps of x0 each? So x1 + x2 = a*x0 + b*x0 where a and b are different positive integers. Now you have a series .of equal-length steps. And you know that equal-length steps of any number give a zero mean, right?

 

[hutchphd]

You don't need any fancy math.
Say you have two steps of different lengths x1 and x2. How about dividing each of those into smaller, equal-length steps of x0 each? So x1 + x2 = a*x0 + b*x0 where a and b are different positive integers. Now you have a series .of equal-length steps. And you know that equal-length steps of any number give a zero mean, right?

Why would you do this? The other result is two lines and it is patently correct. And the "fancy math" is the common language of probability which is the pedagogic purpose here.

 

[arcTomato]

I think I got it
How can I write the calculation process?

$<k>=<k_0+k_1+k_2,,,,>=<k_0>+<k_1>+<k_2>+,,,,,=0$
right??

 

[hutchphd]

I think I got it
How can I write the calculation process?

$<k>=<k_0+k_1+k_2,,,,>=<k_0>+<k_1>+<k_2>+,,,,,=0$
right??

Absolutely.
I will let you worry with the formalism for a while...work it from the end backwards if it helps. Or meet in the middle.

 

[arcTomato]

I appreciate for your help!
Tell me again if there is a problem I do not understand!

 

[haruspex]

You don't need any fancy math.
Say you have two steps of different lengths x1 and x2. How about dividing each of those into smaller, equal-length steps of x0 each? So x1 + x2 = a*x0 + b*x0 where a and b are different positive integers. Now you have a series .of equal-length steps. And you know that equal-length steps of any number give a zero mean, right?

I don't see how that results in a proof. What is the mapping between this and the original problem? E.g. how does a sequence of n steps translate, and where might you end up?
Secondly, it gets a little messy if the two lengths have an irrational ratio.
Thirdly, although I do not get the point of the hint, you seem to have misread it. It says, a succession of random walks, wherein each random walk has a single step length (implying that different walks in the sequence can have different step lengths).