Variance of A sin(Fx)+Bx, and correlation

I was wondering about the statistical variance of $$A\sin(F x)+ Bx$$, and the covariance (or correlation) between $$A_{1} \sin(F_{1} x) + B_{1} x$$ and $$A_{2} \sin(F_{2} x) + B_{2} x$$, assuming that $$x$$ is uniformly distributed in $$[0, U]$$, and $$A, A_1, A_2, F, F_1, F_2, B, U$$ are constants.

How can I proceed to compute these quantities in the simplest way?

• $X$ is uniform where? We need an interval to start. Commented May 9, 2023 at 15:03
• uniform on [0, U]
– Jada
Commented May 9, 2023 at 15:32
• Have you tried writing down the definition of the variance/covariance? That should give you an integral, right? Have you tried to evaluate the integral? Commented May 9, 2023 at 15:48
• I have not succeeded, as the definition seemed difficult to apply: hoping in some simpler way. Note that there is probably no need to refer to a distribution, it could be simply a Lebesgue measure.
– Jada
Commented May 9, 2023 at 15:51
• At an intuitive level, I would expect the correlation to be close to the geometric mean of the B's.
– Jada
Commented May 9, 2023 at 16:01

1 Answer

I'm going to simplify your notation some. I will have capital letters refer to random variables and lowercase letters refer to the parameters. To that end, suppose $$X \sim \mathcal{U}(0, u)$$ (that is, $$X$$ follows a uniform distribution on $$[0, u]$$). Consider the random variable $$Y = a\sin(\lambda X) + bX.$$ I'll leave it to you to compute the variance, knowing that $$\sigma_X ^2 = E(X^2) - (E(X))^2$$

(hint: integrate by parts). Once you have the variance, now use the correlation formula $$\rho_{X, Y} = \frac{\mathrm{Cov}(X, Y)}{\sigma_X \sigma_Y}$$

where the covariance is a double integral in this case, i.e.

$$\mathrm{Cov}(X, Y) =E\left[ (X - E(X))(Y - E(Y) \right].$$

This, again, may require parts.

• Yes, thank you. With your formula above, the variance seems easier to compute (with respect to the classic definition). I need to see how to do the correlation which seems a bit more complicated :-) Perhaps a similar simplification can be done for that too by using mixed moments ...
– Jada
Commented May 9, 2023 at 20:37