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 $E[X_sX_t]$


There is no need to be confused, just calculate the given expectations by using independence:

$$E[X_t] = E\left[A\sin(\omega t + B)\right] = E[A]\cdot E\left[\sin(\omega t + B)\right] = \frac{1}{\lambda} \cdot 0 = 0$$ 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 $a,b\in\Bbb R$


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