Answer to Question #225805 in Statistics and Probability for Prathyush

Question

(a) Give two examples of continuous time discrete state random processes.
(b) Find the mean, autocorrelation and autocovariance of the random process defined by X(t) =
A cos ωt, where A is uniformly distributed in (0,1) and ω is a constant.

Answer 1

Part a)

Discrete-time continuous state Markov processes are widely used. Autoregressive processes are a very important example.

Actually, if you relax the Markov property and look at discrete-time continuous state stochastic processes in general, then this is the topic of study of a huge part of Time series analysis and signal processing.

The most famous examples are ARMA processes, the Conditionally Heteroscedastic models, a large subclass of Hidden Markov models

Part b)

Given, random process, $X(t) = Kcos(wt) t ≥ 0$

The expected value of K is, $E[K]=\frac{1+0}{2}=0.5$

The variance of K is, $\sigma_K^2 = \frac{(1-0)^2}{12}=\frac{1}{12}$

Thus

$E[K^2] = \sigma_K^2+ [E[K]]^2 \\ E[K^2] = \frac{1}{12} + (0.5)^2 \\ E[K^2] = \frac{1}{ 3}$

The mean of X(t) is,

$E[X(t)] = E[Kcos(wt)] \\ E[X(t)] = E[K]cos(wt) \\ E[X(t)] = \frac{1}{2} cos(wt)$

The autocorrelation function of X(t) is,

$R_{xx}(t, s) = E[X(t)X(s)] \\ R_{xx}(t, s) = E[K^2cos(wt)cos(ws)] \\ R_{xx}(t, s) = E[K^2]cos(wt)cos(ws)\\ R_{xx}(t, s) = \frac{1}{3} cos(wt)cos(ws)$

The autocovariance function of X(t) is,

$C_{xx}(t, s) = R_{xx}(t, s) — E[X(t)]E[X (s)]\\ C_{xx}(t, s) = \frac{1}{3}cos(wt)cos(ws) -\frac{1}{4} cos(wt)cos(ws)\\ C_{xx}(t, s) = \frac{1}{12}cos(wt)cos(ws)$