# Solution {-}

A stationary Gaussian process $X(t)$ has power spectral density function:

\begin{equation*}
  S_X(j\omega)=\frac{2}{\omega^4 + 1}
\end{equation*}

1. Find $E(X)$ and $E(X^2)$


```python
from numpy import linspace
import matplotlib.pyplot as plt

omega = linspace(-5, 5, 100)
Sx = 2/(omega**4 + 1)

plt.plot(omega, Sx)
plt.title("Spectral density")
plt.xlabel("Frequency")
plt.ylabel("Power")
plt.grid()
plt.show()
```


```python
from sympy import symbols, integrate, pi, oo

omega = symbols('omega')

Sx = 2/(omega**4 + 1)

EX = 0 # No impulse function at the origin of Sx
display(EX)

EX2 = 1/(2*pi)*integrate(Sx, (omega, -oo, oo))
display(EX2)
```


    0



$\displaystyle \frac{\sqrt{2}}{2}$


974d243e47c5d95332a7e0ec60d2a54c0af75d44
ipynb
Jupyter Notebook
Problem 2.18.ipynb
mfkiwl/GMPE340
3602b8ba859a2c7db2cab96862472597dc1ac793
MIT
2021-09-07T09:36:36.000Z
2021-09-07T09:36:36.000Z
Problem 2.18.ipynb
mfkiwl/GMPE340
3602b8ba859a2c7db2cab96862472597dc1ac793
MIT
Problem 2.18.ipynb
mfkiwl/GMPE340
3602b8ba859a2c7db2cab96862472597dc1ac793
MIT
2021-09-20T18:48:20.000Z
2021-09-20T18:48:20.000Z
Qwen/Qwen-72B
1. YES
2. YES