Exercise :

Consider two independent random variables $A,Θ$ and define the stochastic process $\{ X_t \}_{t \geq 0}$ with the formula : $$X_t = A\sin(ωt + Θ)$$ where $ω \in \mathbb R$. If the random variable $A$ follows an exponential distribution with $λ=1$ and $Θ$ a uniform distribution in $[0,2\pi]$, compute the $E[X_t]$ and $E[X_t X_s]$, for all $s,t \geq 0$.

Attempt - Discussion :

Starting off determining the mean and variance of the random variables defined :

$$E[A] = 1/λ = 1, \space V[A] = 1/λ^2 = 1$$

$$E[Θ] = \frac{1}{2}(0+2\pi)=\pi, \space V[Θ] = \frac{1}{12}(b-a)^2=\frac{1}{3}\pi^2$$

Now, in this case that this is a joint distribution example (considering that the expression for the stochastic process has two differently distributed random variables), how should I approach the computations of $E[X_t]$ and $E[X_tX_s]$ ? Sorry if this is a soft question but I'm a early beginner on Stochastic Processes.

  • 1
    $\begingroup$ You might start by proving that, for every $u$, $$E(\cos(u+\Theta))=E(\sin(u+\Theta))=0$$ since the rest follows easily. $\endgroup$ – Did Mar 5 '18 at 20:17
  • $\begingroup$ @Did Thanks for the quick reply ! I'll start so ! Any recommendations on where I could find rigid examples and good exercises ? $\endgroup$ – Rebellos Mar 5 '18 at 20:19

$$E(X_t)=\int_{0}^{\infty}\int_{0}^{2\pi}e^{-a}\dfrac{1}{2\pi}a\sin(\omega t+\phi)d\phi da=0$$also $$E(X_tX_s)=\int_{0}^{\infty}\int_{0}^{2\pi}e^{-a}\dfrac{1}{2\pi}a^2\sin(\omega t+\phi)\sin(\omega s+\phi)d\phi da=\dfrac{1}{2}\cos\omega(s-t)\int_{0}^{\infty}a^2e^{-a}da=\cos\omega(s-t)\qquad,\qquad s,t\ge0$$


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