How to show the root mean square deviation

In summary, the conversation discusses the calculation of 蟽^2 in the context of random walks in one dimension. The formula for 蟽^2 is given and the speaker is trying to understand it, asking for clarification on why 位鈮1. The other person explains that a random variable is needed and provides an example to demonstrate this, ultimately leading to the conclusion that the formula for 蟽^2 is proportional to the number of steps taken.
  • #1
arcTomato
105
27
Homework Statement
prove that the mean position after a given number of steps is the starting position
Relevant Equations
random walk
246565

Hi
I tried like this.
##蟽^2=<(位_1+位_2+,,,+位_i)^2>=<位_1^2>+<位_2^2>+,,,<位_i^2>##
And I know ##蟽^2=危_in_i位_i^2##from equation (4-12) (so this is cheat 馃槄).

So I know also ##<位_i^2>=n_i位_i^2##, But why??

I know if I take ##位=1 ,蟽^2=n##,But I don't understand ##位鈮1## version.

Sorry my bad english.
Please help me!
 
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  • #2
arcTomato said:
I tried like this.
##蟽^2=<(位_1+位_2+...+位_i)^2>=<位_1^2>+<位_2^2>+...+<位_i^2>##

Why do you think that is true? If I understand the problem correctly, the ##\lambda_i## are fixed parameters and so are the ##n_i##. You need a random variable here.

Are you studying random walks in one dimension? Then let's define a random variable ##\Lambda_{ij}##, ##j = 1,n_i## which is either ##\lambda_i## or ##-\lambda_i## with equal probability. That is, corresponding to each ##\lambda_i## (fixed) there are ##n_i## (fixed number) of steps, each of which is either ##\lambda_i## to the right or ##\lambda_i## to the left (randomly).

Let's start with a simpler example. Let's suppose all the steps have the same size ##\lambda_1## and there are ##n_1## of them. So ##N = n_1##. So we have ##\Lambda_{11}, \Lambda_{12}, ..., \Lambda_{1,n_1} = \pm\lambda_1##. Obviously each has zero mean.

The total square deviation we are interested in is then ##\sigma^2 = <\left ( \sum_{j=1}^{n_1} \Lambda_{1j} \right )^2 > = \sum_{j=1}^{n_1} < \Lambda_{1j}^2 >## since the ##\Lambda_{1j}## are independent and the expectation of the cross terms is 0. Thus ##\sigma^2## in this case reduces to ##n_1 <\Lambda_{11}^2>##.

This is the familiar result that ##\sigma^2## is proportional to the number of steps. The final thing here (I leave it to you) is to replace the expectation value of ##\Lambda_{11}^2## by something in terms of the fixed parameters ##\lambda##.

You may think that this is what you were doing, but it's not. You didn't have a random variable in your expression, and you didn't have the right number of terms. You need to have a random variable for each of the ##n_i## steps of each size, as I did even when there is only one step size. So your steps will be ##\Lambda_{11}, \Lambda_{12}, ..., \Lambda_{1n_1} = \pm\lambda_1## and ##\Lambda_{21}, \Lambda_{22}, ..., \Lambda_{2n_2} = \pm \lambda_2##, etc., and you will need double sums.
 
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  • #3
At first, Thank you for reply. Your explanation is so easy to understand.
I think I didn't understand the problem correctly.

I have to take the length like##螞_{i,1},螞_{i,2}...,螞_{i,n_i}=卤位_i##.
so
##\sigma^2 = <\left ( \sum_{j=1}^{n_i}\sum_{i} \Lambda_{ij} \right )^2 > = \sum_{i} \sum_{j=1}^{n_i} < \Lambda_{ij}^2 >=\sum_{i} n_i(卤\lambda_{i})^2 ##
I think this is almost the answer. Right??
 
Last edited:

Related to How to show the root mean square deviation

1. What is the root mean square deviation (RMSD)?

The root mean square deviation, also known as root mean square error, is a statistical measure of the differences between a set of values and their corresponding expected values. It is commonly used to assess the accuracy of a model or to compare different sets of data.

2. How is the RMSD calculated?

The RMSD is calculated by taking the square root of the average of the squared differences between each value and its expected value. This value is then used to measure the overall deviation from the expected values.

3. What is the significance of the RMSD?

The RMSD is an important measure in the field of statistics as it provides a way to evaluate the accuracy of a model or the consistency of a set of data. It is often used in fields such as chemistry, physics, and engineering to assess the differences between experimental and theoretical values.

4. How is the RMSD interpreted?

The RMSD is typically interpreted in the same unit as the values being measured. A lower RMSD indicates a better fit between the observed and expected values, while a higher RMSD indicates a larger deviation from the expected values.

5. Can the RMSD be used for all types of data?

The RMSD can be used for any type of data as long as there is a clear expected value to compare it to. It is commonly used in regression analysis, time series analysis, and other statistical methods to evaluate the accuracy of a model or the consistency of a set of data.

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