Mean value of random discrete signal

Question

We have the following discrete signal $$X(n)=A\cos(ω_0n+Φ)$$with probability density function for $A,Φ$ $$ ρ_A(a)=1/4$$for $0\le a\le4$ and $$ρ_Φ(φ)=1/2π$$ for $0\le φ\le 2π$

I found the joint probability density function $ρ_{A,Φ}(α,φ)=1/8π$ for $0\le a\le 4$ and $0\le φ \le 2π$ . How do I proceed to find the mean value of $X(n)$? I normally integrate $aρ_{X(n)}$ but I don't know what to put inside the integral in this problem.

Answer 1

From what I understand $X(n)$ is a function of the random variables $A$ and $\Phi$.

Then $$\mathbb E [X(n)] = \int a \cos(\omega_0n+\phi) \rho(a, \phi) \, \mathrm{d}a \, \mathrm{d}\phi $$

You obtain $$\mathbb E [X(n)] = \int_0^4 a \rho_A(a)\,\mathrm{d}a \int_0^{2\pi} \cos(\omega_0n+\phi) \rho_\Phi(\phi) \, \mathrm{d}\phi = 0 $$