Are the Mean Values of Two Variables Related by a Square Root?

[Niles]

Homework Statement

Hi all.

Lets say I have two variables y_k and x_k. I also have two mean values given by:

$$<y_k^2> = \frac{1}{N}\sum_1^N{y_k^2} \quad \text{and} \quad <x_k^4> = \frac{1}{N}\sum_1^N{x_k^4}.$$

Now I am looking at the expression (<x_k⁴> <y_k²>)^1/2.

Question: Is it correct that:

$$<y_kx_k^2> = \sqrt{<x_k^4><y_k^2>}.$$

Personally, I don't think so, because ultimately it would mean that I would have to make two sums into one sum, which I can't.. but I am in doubt.

Thanks in advance.

Niles.

 

[Mark44]

Per your notation (< ... >), <y_k²> denotes the mean of the squared values of y_k.

So <x_k⁴y_k²> would be the sum of the products of x_k⁴y_k², divided by N, which is not the same as (<x_k⁴> <y_k²>)^1/2.
The latter would just be the square root of (the mean of the x^4 terms times the mean of the y^2 terms).

 

[Niles]

Yeah, just what I thought.. so they are not the same.

Thanks.