I am working on a problem and hoping to receive either confirmation that my work is correct or guidance on where I am going wrong. The problem is shown below;
Here is my work;
a) $$ E\{ X(t) \} = \eta_{_X} (t) = 2 E\{ Z(t) \} + 3 \lambda = 2 \lambda + 3 \lambda = 5 \lambda \\ E\{ Y(t) \} = \int_{-\infty}^\infty \eta_{_X} (t - \tau) h(\tau) d\tau = 5 \lambda \int_{0}^\infty e^{-a \tau} d\tau = \frac{5 \lambda}{a} $$
where $ E\{ Z(t) \} = \lambda $ is given in the problem statement.
b)
$$ \begin{align} R_{_{XX}}(\tau) = E\{ X(\tau)^2 \} &= E\{ (2 Z(\tau) + 3 \lambda)^2 \} \\ &= 4 E\{ Z (\tau)^2 \} + 12 \lambda E\{ Z (\tau) \} + 9 \lambda^2 \\ &= 4 R_{_{ZZ}}(\tau) + 12 \lambda E\{ Z (\tau) \} + 9 \lambda^2 \\ &= 4\left( \lambda^2 + \lambda \delta(\tau) \right) + 12 \lambda(\lambda) + 9 \lambda^2 \\ &= 25 \lambda^2 + 4 \lambda \delta(\tau) \end{align} $$ where $ R_{_{ZZ}}(\tau) = \lambda^2 + \lambda \delta(\tau) $ is given in the problem statement and $ \tau = t_2 - t_1 $. Next, we take the Fourier transform of $ R_{_{XX}}(\tau) $ to get the spectral density ($S_{_{XX}}(\omega)$). Additionally, we take the fourier transform of the transfer function $ h(t) $.
$$ S_{_{XX}}(\omega) = \mathscr{F} \{ R_{_{XX}}(\tau) \} = \mathscr{F} \{ 25 \lambda^2 + 4 \lambda \delta(\tau) \} = 25 \lambda^2 \delta(\omega) + 4 \lambda \\ H(\omega) = \mathscr{F} \{ h(\tau) \} = \mathscr{F} \{ e^{-a \tau} u(\tau) \} = \frac{1}{a + i \omega} $$
where $i = \sqrt{-1} $. Finally, we obtain the power spectral density of the output ($S_{_{YY}}(\omega)$) as follows;
$$ S_{_{YY}}(\omega) = S_{_{XX}}(\omega)H(\omega)H(-\omega) = \frac{25 \lambda^2 \delta(\omega) + 4 \lambda}{(a + i \omega)(a - i \omega) } = \frac{25 \lambda^2 \delta(\omega) + 4 \lambda}{ a^2 + \omega^2 } $$
Is this work correct? Any comments/corrections are greatly appreciated.
