Expectation of a Function of a RV in terms of PDF

[RoshanBBQ]

Homework Statement

Find the expected value of cos(A+B) where A is a constant and B is a random variable with a pdf f(b). Present the answer in terms of f(b).

The Attempt at a Solution

I don't know how far I can go with the answer -- I have tried for a bit now to remove an integral with no success.

$$\int\limits_{-\infty}^{\infty}\cos(A+b)f(b)\, db$$

From here, I tried integration by parts, which ended in evaluating a cosine at infinity (for the uv term). I thought perhaps to find the pdf of the r.v. C = cos(A + B) and then integrate cf_c(c), but the issue of removing the integral seems strong as ever with that approach too.

Do you think the teacher intends for the answer to have an integral in it?

 

[Ray Vickson]

Homework Statement

Find the expected value of cos(A+B) where A is a constant and B is a random variable with a pdf f(b). Present the answer in terms of f(b).

The Attempt at a Solution

I don't know how far I can go with the answer -- I have tried for a bit now to remove an integral with no success.

$$\int\limits_{-\infty}^{\infty}\cos(A+b)f(b)\, db$$

From here, I tried integration by parts, which ended in evaluating a cosine at infinity (for the uv term). I thought perhaps to find the pdf of the r.v. C = cos(A + B) and then integrate cf_c(c), but the issue of removing the integral seems strong as ever with that approach too.

Do you think the teacher intends for the answer to have an integral in it?

Yes, of course. There is no other way to do it.

RGV