Let {${X(t); t\in ℝ}$} be a wide sense stationary Gaussian process with mean $\mu_X = 1$ and power spectral density $$S_X(f) = \begin{cases} 1, \ \text{if} \ |f| < 5; \\ 0, \ \text{otherwise}. \end{cases}$$
This process is the input of a LTI with impulse response $$h(t) = \begin{cases} 8e^{-4t}, \ \text{if} \ t >0; \\0,\ \ \ \ \ \ \ \text{otherwise}. \end{cases}$$
Let {${Y(t); t\in ℝ}$} be the output signal.
We must compute $P(Y(1)>1)$.
As {${X(t); t\in ℝ}$} is wide sense stationary, I know that we can compute its autocorrelation function as the inverse Fourier transform of its power spectral density $S_X(f)$. This gives us : $$ R_X(\tau) = sinc \ (\frac {5} {\pi} \tau)$$
Here is my question, I don't know how to make the link between this autocorrelation function and the probability that we want to compute. Should I compute the convolution $$ R_X(\tau) \ \ * \ \ h(1)$$
But then I don't how to use the given mean...
Or should I use the fact that the input signal {${X(t); t\in ℝ}$} is a Gaussian process to say that the output signal also is, and somehow compute the probability using this ?
Does anyone have an idea ? Thanks.