# $Ε[Χ]$ calculations with different distribution variables

I have just started my stochastic processes course this semester and I came across this problem. I am missing something for sure because the $$X_s$$ confuses me.

Let $$A,B$$ two independent random variables and $$\{X_t\}_{t\ge0}$$ a stochastic process, where $$X_t=A\sin(\omega t+B)$$ for some $$\omega\in\mathbb R$$.

If $$A\sim Exp(λ)$$ with $$\lambda=1$$ and $$B\sim U(0, 2\pi)$$, calculate $$E[X_t]$$ and $$E[X_sX_t]$$

• – user190080 Mar 3 at 13:03

$$E[X_t] = E\left[A\sin(\omega t + B)\right] = E[A]\cdot E\left[\sin(\omega t + B)\right] = \frac{1}{\lambda} \cdot 0 = 0$$ and \begin{align*}E[X_sX_t] &= E[A\sin(\omega t + B)\cdot A\sin(\omega s + B)] \\ &= E[A^2] \cdot E\left[\sin(\omega t + B)\sin(\omega s + B)\right] \\ &= \frac{2}{\lambda^2} \cdot \frac{1}{2}\cos(\omega (t-s)) \\\\&= \frac{\cos(\omega (t-s))}{\lambda^2}\end{align*}
where for the next to last equation it was used that $$\text{Var}(A) = E[A^2] - E^2[A]$$ and
$$\frac{d}{dx} \left(\frac{1}{2}x\cos(a - b) - \frac{1}{4}\sin(a + b + 2 x)\right) = \sin(a+x)\sin(b+x)$$ for arbitraty $$a,b\in\Bbb R$$