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.