• 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.