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 ${\sigma }_X^2(t)$ 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 $$C_X(t,s)=\text{Cov}(X(t),X(s))=\text{E}[X(t)X(s)]-\text{E}[X(t)]\text{E}[X(s)]$$
• 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 $$\begin{array}{rl}{C}_X(t,s)& =R_X(t,s)-{\mu }_X(t){\mu }_X(s)\\ {\sigma }_X^2(t)& =C_X(t,t)\end{array}$$

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 $X(t)=\cos (2\pi t+\mathrm{\Theta })$, where the random phase shift $\mathrm{\Theta }$ 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.