Variance of 2-D random walker

[ragnabob]

I've made a 2D walker to compare different RNG's. I'm measuring the succes of each walk as the distance from the origin to the endpoint, using the regular 2-norm. The thing I can't seem to work out is the variance.

$$D_n=\sqrt(x_n^2+y_n^2)$$

$$Var(D_n)=E[D_n^2]=E[Z_1^2+...+Z_n^2]$$

Since $Var(Z_i)=\sqrt{2}$ does this mean that the variance is $2n$? Seems too easy...

Ps. I'm not sure how to make the formatting prettier, if someone can tell me, I'll edit it naturally!
Ps2. Thanks Stephen Tashi!

 

[Stephen Tashi]

Ps. I'm not sure how to make the formatting prettier, if someone can tell me, I'll edit it naturally!

On this forum, surround the LaTex with "tags" rather than the dollar sign.

[tex]D_n=\sqrt{x_n^2+y_n^2}[/tex]

[tex]Var(D_n)=E[D_n^2]=E[Z_1^2+...+Z_n^2][/tex]

Since [itex]Var(Z_i)=\sqrt{2}[/itex] does this mean that the variance is [itex]2n[/itex] ?

$$D_n=\sqrt{x_n^2+y_n^2}$$

$$Var(D_n)=E[D_n^2]=E[Z_1^2+...+Z_n^2]$$

Since $Var(Z_i)=\sqrt{2}$ does this mean that the variance is $2n$ ?

 

[Stephen Tashi]

Another thing about the forum: When you edit a post, sometimes "Save" doesn't display the LaTex. You must refresh the page to accomplish that.

$Var(z_i) = \sqrt{2}$ for the random variable $z_i$ that uses the square of the distance between the current position and the previous position. But this is not the same as using the distance between the current position and $(x_0,y_0)$.

For example, there is the distinction between
$$z_2 = \sqrt{(x_1-x_0)^2 + (y_1-y_0)^2} + \sqrt{( x_2-x_1)^2 + (y_2-y_1)^2}$$

and

$$Z_2 = \sqrt{ (x_2-x_0)^2 + (y_2-y_0)^2}$$

 

[PatrickPowers]

I've made a 2D walker to compare different RNG's. I'm measuring the succes of each walk as the distance from the origin to the endpoint, using the regular 2-norm. The thing I can't seem to work out is the variance.

$$D_n=\sqrt(x_n^2+y_n^2)$$

$$Var(D_n)=E[D_n^2]=E[Z_1^2+...+Z_n^2]$$

Since $Var(Z_i)=\sqrt{2}$ does this mean that the variance is $2n$? Seems too easy...

Ps. I'm not sure how to make the formatting prettier, if someone can tell me, I'll edit it naturally!
Ps2. Thanks Stephen Tashi!

The sum of the variances of independent random variables is the variance of the sum. That should make it easy, unless I'm missing something.

 

[blue_raver22]

I would first like to commend you on your approach to comparing different random number generators (RNGs). Using a 2D walker and measuring the success based on the distance from the origin to the endpoint is a great way to evaluate the performance of each RNG.

Now, regarding your question about the variance, I would like to clarify a few things. First, the variance of a random variable is a measure of how spread out the values are from the average or expected value. In your case, the random variable is the distance from the origin to the endpoint, and the expected value is the average distance after n steps.

Second, the formula you have for the variance, Var(D_n) = E[D_n^2] = E[Z_1^2+...+Z_n^2], is correct. This means that the variance is equal to the expected value of the squared distance, which is the sum of squares of the individual distances for each step.

However, the next statement, "Since Var(Z_i)=√2 does this mean that the variance is 2n?" is not entirely accurate. The square root of 2 (√2) is the standard deviation of the random variable Z_i, which represents the distance for each individual step. The variance is actually equal to the sum of the squared standard deviations, i.e. Var(Z_i) = (√2)^2 = 2.

Therefore, the correct formula for the variance in your case would be Var(D_n) = 2n. This means that the variance of the distance after n steps is equal to 2 times the number of steps. This makes sense because as the number of steps increases, the distance from the origin to the endpoint also increases, resulting in a larger spread of values and thus a higher variance.

In conclusion, your approach to comparing RNGs using a 2D walker is valid and the formula for the variance is Var(D_n) = 2n. I hope this clarifies your question and helps you in your research. As for formatting, I would recommend using LaTeX for mathematical expressions to make it easier to read and understand. Thank you for sharing your work and good luck with your research!