Random function coupled to a non-random function question.

KayBox

Hello and thank you in advance for anyone taking time to respond.

I working on formulating a theory for elastodynamics, but my statistics is admittedly weak. I'm trying to find a relationship between a non random function and a random function, for example, the covariance.

<A(x)B(y)>=some 2 point probability for 2 random functions.

These are discrete random variables, so the result is usually calculated numerically or by experimental observation.

The problem I'm having is what if B(y) is non-random (ie deterministic) and continuous (not discrete)? I understand there is a discrete and integral formulation for discrete and continuous functions respectively, but I keep getting stuck. Its like I'm trying to find a connection that shouldn't be there, but they always come up in the formulation. Can anyone give me an idea of where to start?

mathman

If B(y) is not random, then <A(x)B(y)> = B(y)<A(x)>.

KayBox

Ah, not what I wanted to hear (<A(x)>=0 centered fluctuations).

What if B(y) is a binary characteristic function that is 1 inside a specified volume size within the total volume(though its position is not known) and zero otherwise?

mathman

You need to be specific about the relationship between y and x.

KayBox

y and x are both chosen at random. When A(x) and B(y) correspond to random material property fluctuations, the correlation function is either assumed to be of the form exp(r/L) where r is the absolute distance between the points and L is the average grain size for the polycrystalline material, or is evaluated numerically. When evaluated numerically using a Voronoi tesselation, randomly tossing line segments in the model, then calculating the product produces the numerical result, which is then fitted to some function similar to the assumed form above. Its a measure of the likelihood that the two points are in the same grain.

For this case, x and y are still chosen at random, but only one varies randomly in space. The other is deterministic, in that its center is located in the inspection volume with a its own volume occupied by the foreign material, while the other varies randomly. Its basically an inclusion type problem in a polycrystalline medium. Hope this helps.

mathman

Since y is random B(y) is random (even if B is deterministic), so you cannot take it outside the bracket.

KayBox

Thanks for your help. Looks like I will have to determine the correlation functions numerically.