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

• arcTomato
In summary, the problem is trying to find a sequence of steps that will take you to a certain position, but with a given number of steps the final position will be the same as the starting position.
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,,,

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??

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

arcTomato
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
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 :<

arcTomato said:
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
hutchphd said:
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. ;;

arcTomato said:
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??

arcTomato said:
##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

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?

rude man said:
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.

sysprog
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??

arcTomato said:
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.

Tell me again if there is a problem I do not understand!

rude man said:
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).

1. How do I approach solving a random walk problem?

Solving a random walk problem involves understanding the basic principles of random walks and then applying them to the specific problem at hand. It can be helpful to break down the problem into smaller, more manageable steps and consider different scenarios or variations of the problem.

2. What mathematical concepts are needed to prove a random walk problem?

To prove a random walk problem, you will need a solid understanding of probability, statistics, and basic algebra. You may also need to use concepts from calculus, such as limits and derivatives, depending on the complexity of the problem.

3. How can I visualize a random walk problem?

Visualizing a random walk problem can be helpful in understanding the problem and developing a proof. One way to do this is by drawing a graph or diagram to represent the steps or movements of the random walk. You can also use computer simulations or programming to visualize and analyze the problem.

4. Are there any common strategies or techniques for proving random walk problems?

Yes, there are several common strategies and techniques that can be used to prove random walk problems. These include using symmetry arguments, considering boundary conditions, and using mathematical induction. It can also be helpful to look for patterns or relationships within the problem and use them to develop a proof.

5. What are some common mistakes to avoid when proving a random walk problem?

One common mistake when proving a random walk problem is assuming that the problem has a simple or straightforward solution. It is important to carefully consider all possibilities and variations of the problem before attempting to prove a solution. It is also important to clearly define and explain all steps in the proof, and to check for any errors or inconsistencies in your reasoning.

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