30  Mean and Autocorrelation Functions of Stochastic Processes

• Recall that a realization of a stochastic process \(\{X(t),t\ge 0\}\) is a sample path (function of \(t\)).
• At any given point in time \(t\), \(X(t)\) is a random variable
• So we can compute the expected value (a.k.a. mean) of \(X(t)\), \(\mu_{X}(t)=\text{E}[X(t)]\)
• At a given point in time \(t\), \(\mu_X(t)\) is a number
• The (ensemble) mean function of a stochastic process \(\{X(t)\}\) is the function \(\mu_X(t)\) which maps \(t\mapsto \text{E}[X(t)]\).
• Because for each \(t\) the quantity \(\mu_X(t)\) is a number, the mean function \(\mu_X(t)\) is a deterministic (i.e., not random) function of time.
• At a given time \(t\), \(\mu_X(t)\) represents the average value of the process at time \(t\), \(X(t)\), over all the sample paths in the ensemble.
• Similarly, the (ensemble) variance function of a stochastic process \(\{X(t)\}\) is the function \(\sigma^2_X(t)\) which maps \(t\mapsto \text{Var}[X(t)]\).

Example 30.1 Harry and Tom play a game which involves flipping a fair coin. Each time the coin lands on heads, Tom pays Harry 1 dollar; each time the coin lands on tails, Harry pays Tom 1 dollar. Let \(X_n\) denote Harry’s cumulative winnings after \(n\) flips, with \(X_0=0\), and with negative values denoting losses.

1. Find the mean function.
   
   
   
   
   

2. Find the variance function.
   
   
   
   
   

Example 30.2 Consider the stochastic process \(X(t) =A \cos(2\pi t)\), where the amplitude variation \(A\) has an Exponential(0.5) distribution.

1. Evaluate the mean function at \(t=0.5\).
   
   
   
   
   

2. Find the mean function.
   
   
   
   
   

3. Evaluate the variance function at \(t=0.5\).
   
   
   
   
   

4. Find the variance function.
   
   
   
   
   

Example 30.3 Consider the stochastic process \(X(t) = s(t) + N(t)\), where \(s(t)\) is a non-random signal (e.g., \(s(t)=3\cos(2\pi(100) t)\)) and \(N(t)\) is random noise, for which at any time \(t\), \(N(t)\) has a Gaussian distribution with mean 0 and standard deviation 2.

1. Evaluate the mean function at \(t=1\).
   
   
   
   
   

2. Find the mean function.
   
   
   
   
   

3. Evaluate the variance function at \(t=1\).
   
   
   
   
   

4. Find the variance function.
   
   
   
   
   

• In a stochastic process \(X(t)\) represents the value at time \(t\) for some random process which occurs over time.
• At each fixed point in time \(t\), \(X(t)\) is a random variable, and so it has a probability distribution.
• At any two points in time, \(t\) and \(s\), both \(X(t)\) and \(X(s)\) are random variables, so they have a joint distribution.
• So we can compute their covariance \[ C_{X}(t,s) = \text{Cov}(X(t),X(s)) = \text{E}\left[X(t)X(s)\right]-\text{E}\left[X(t)\right]\text{E}\left[X(s)\right] \]
• The (ensemble) autocovariance function of a stochastic process \(\{X(t)\}\) is the function \(C_{X}(t,s)\) which maps \((t,s)\mapsto \text{Cov}[X(t),X(s)]\).
• Because for each \((t,s)\) pair the value \(C_{X}(t,s)\) is a number, the autocovariance function is a deterministic (i.e., not random) function of two time points.
• The (ensemble) autocorrelation function of a stochastic process \(\{X(t)\}\) is the function \(R_{X}(t,s)\) which maps \((t,s)\mapsto \text{E}[X(t)X(s)]\).
  • In engineering applications, the correlation of two random variables \(X\) and \(Y\) is defined to be \(\text{E}(XY)\).
  • Warning! Do not confuse engineering correlation with the statistical correlation coefficient, \(\frac{\text{Cov}(X,Y)}{\text{SD}(X)\text{SD}(Y)}\)
• Relationships \[\begin{align*} C_{X}(t, s) & = R_{X}(t,s) - \mu_X(t)\mu_X(s)\\ \sigma^2_X(t) & = C_{X}(t,t) \end{align*}\]

Example 30.4 Let \(X(t) = A \cos(2\pi t)\), where the amplitude variation \(A\) has an Exponential(0.5) distribution.

1. Evaluate the autocorrelation function at \(t = 0.5\) and \(s=2/3\).
   
   
   
   
   

2. Find the autocorrelation function.
   
   
   
   
   

3. Find the autocovariance function.
   
   
   
   
   

4. For \(t=0\), plot the autocovariance function as a function of \(s\). Explain why the plot makes sense.
   
   
   
   
   

Example 30.5 In radio communications, the carrier signal is often modeled as a sinusoid with a random phase. (The reason for using a random phase is that the receiver does not know the time when the transmitter was turned on or the distance from the transmitter to the receiver.) Consider the stochastic process \(X(t) = \cos(2\pi t + \Theta)\), where the random phase shift \(\Theta\) has a Uniform\((0, 2\pi)\) distribution.

1. Find the mean of \(X(1)\).
   
   
   
   
   

2. Find the mean function.
   
   
   
   
   

3. Evaluate the autocorrelation function at \(t = 1\) and \(s=1.2\).
   
   
   
   
   

4. Find the autocorrelation function.
   
   
   
   
   
   

5. Find the autocovariance function.
   
   
   
   
   

6. Find the variance function.