# Solution {-}

The stationary process $X(t)$ has an autocorrelation function of the form:

\begin{equation}
  R_X(t) = \sigma^2 e^{-\beta|\tau|}
\end{equation}

Another process $Y(t)$ is related to $X(t)$ by the deterministic equation:

\begin{equation}
  Y(t) = aX(t) + b
\end{equation}


$X(t)$ is stationary, so Y(t) is also stationary.

\begin{align}
  R_Y(t) =& E[Y(t) \cdot Y (t + \tau)] \\
         =& E[(aX(t) + b)(aX(t + \tau) + b)] \\
         =& E[(a^2 X(t)X(t + \tau)] + E[abX(t)] + E[baX(t)] + E[b^2] \\
         =& a^2 R_X(t) + b^2 \\
         =& a^2 \sigma^2 e^{-\beta|\tau|} + b^2 \\
\end{align}

9f2b713c348b8b2e7b318cb2763f9bf2626e8a61
ipynb
Jupyter Notebook
Problem 2.12.ipynb
mfkiwl/GMPE340
3602b8ba859a2c7db2cab96862472597dc1ac793
MIT
2021-09-07T09:36:36.000Z
2021-09-07T09:36:36.000Z
Problem 2.12.ipynb
mfkiwl/GMPE340
3602b8ba859a2c7db2cab96862472597dc1ac793
MIT
Problem 2.12.ipynb
mfkiwl/GMPE340
3602b8ba859a2c7db2cab96862472597dc1ac793
MIT
2021-09-20T18:48:20.000Z
2021-09-20T18:48:20.000Z
Qwen/Qwen-72B
1. YES
2. YES