How to Compute the Ensemble Average of a Product of Integrals?

[quasar_4]

Homework Statement

I'm trying to compute something of the form $$\langle \int_a^b{f(x) dx} \int_a^b{f(x)^{\dagger}dx} \rangle$$ where the dagger means complex conjugate and the brackets are ensemble average (f(x) is a statistical quantity). I'm supposed to use the relation that $$\langle f(x) f(x')^{\dagger} \rangle = c*\delta(f-f')$$ where c is some constant.

Homework Equations

$$\langle f(x) f(x')^{\dagger} \rangle = c*\delta(f-f')$$

The Attempt at a Solution

I'm a bit perplexed. I have the function and its complex conjugate, but inside different integrals, which are being multiplied. And the ensemble average of a product isn't the same as the product of ensemble averages, either... is it? I'd be surprised.

I thought maybe I could multiply the entire quantity by an extra f dagger, then somehow use the relation, but it didn't really get me anywhere.

So I have no idea how to use the given relation. Can anyone help??

 

[nrqed]

Homework Statement

I'm trying to compute something of the form $$\langle \int_a^b{f(x) dx} \int_a^b{f(x)^{\dagger}dx} \rangle$$ where the dagger means complex conjugate and the brackets are ensemble average (f(x) is a statistical quantity).

Are you sure that the x appearing in the second integral should not all have a prime on them?

I'm supposed to use the relation that $$\langle f(x) f(x')^{\dagger} \rangle = c*\delta(f-f')$$ where c is some constant.

Are you sure that it is not $\delta(x-x')$ ??


Check these two things and let us know. If I am correct about the two corrections, the problem becomes very easy.

 

[quasar_4]

Oops, you're right. I've been working with functions of frequency and forgot. So yes, should be f(x) and f(x'), and the delta function should then be delta(x-x').

 

[nrqed]

Oops, you're right. I've been working with functions of frequency and forgot. So yes, should be f(x) and f(x'), and the delta function should then be delta(x-x').

Great. Then are you all set? Replacing the product of the functions by a delta function makes the two integrations trivial.

 

[paranormal]

I understand your confusion and I can assure you that this is a common issue in statistical calculations. It seems like you are trying to compute the ensemble average of a product of integrals, which can be quite tricky. However, there are a few ways to approach this problem.

One approach is to use the properties of the delta function, which can be thought of as a generalized function that is zero everywhere except at one point where it is infinite. In this case, the delta function allows us to replace the complex conjugate in the second integral with the original function, but evaluated at a different point. This can simplify the calculation and allow you to use the given relation.

Another approach is to use the properties of ensemble averages. While it is true that the ensemble average of a product is not necessarily equal to the product of ensemble averages, there are certain cases where this holds true. For example, if the two functions are independent of each other, then the product of ensemble averages will be equal to the ensemble average of the product. This may be a useful property to consider in your calculation.

Ultimately, the best approach will depend on the specific details of your problem. I would recommend consulting with your professor or a colleague for further guidance and clarification. Good luck with your calculations!