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Basic Random Processes. Introduction. Annual summer rainfall in Rhode Island is a physical process has been ongoing for all time and will continue. We’d better study the probabilistic characteristics of the rainfall for all times. - PowerPoint PPT Presentation

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Basic Random Processe

Basic Random ProcessesIntroductionAnnual summer rainfall in Rhode Island is a physical process has been ongoing for all time and will continue.Wed better study the probabilistic characteristics of the rainfall for all times.

Let X[n] be a RV that denotes the annual summer rainfall for year n.We will be interested in the behavior of the infinite tuple of RV (, X[-1], X[0], X[1],)IntroductionGiven our interest in the annual summer rainfall, what types of questions are pertinent?A meteorologist might wish to determine if the rainfall totals are increasing with time (there is trend in the data).Assess the probability that the following year the rainfall will be 12 inches or more if we know the entire past history or rainfall totals (prediction).The Korea Composite Stock Price Index or KOSPI () is the index of common stocks traded on the Stock Market Division

What is a random processAssume we toss a coin and the repeat subexperiment at one second intervals for all times.Letting n denoting the time in seconds, we generate outcomes at n = 0,1,.Since there are two possible outcomes a head (X = 1) with probability p and a tail (X = 0) with probability 1 p the processes is termed a Bernoulli RP.S = {(H,H,T,), (H,T,H,), (T,T,H,)}SX = {(1,1,0,), (1,0,1,), (0,0,1,)}Random process generator(x[0],x[1],)(X[0], X[1],)What is a random processEach realization is a sequence of numbers. The set of all realizations is called the ensemble of realizations.

s3s1s2What is a random processThe probability density/mass function describes the general distribution of the magnitude of the random process, but it gives no information on the time or frequency content of the processfX(x)time, t

x(t)Type of Random processes

Discrete-time/discrete-valued(DTDV) Discrete-time/continuous-valued(DTCV) Continuous-time/discrete-valued(CTDV)Continuous-time/continuous-valued(CTCV)

Bernoulli RPGaussian RPBinomial RPGaussian RPRandom WalkLet Ui for i = 1,2,,N be independent RV with the same PMFAt each time n the new RV Xn changes from the old RV Xn-1 by 1 since Xn = Xn-1 + Un.The the joint PMF is

where

Conditional probability of independent eventsand defineRandom WalkNote that can be fond by observing that Xn = Xn-1 + Un and therefore if Xn-1 = xn-1 we have that

Step 1 due toStep due to independenceUns have same PMF

Finally

Realization of UnsRealization of XnsRandom walkThe important property of StationaryThe simplest type of RP is Identically and Independent Distributed (IID) process. (For ex. Bernoulli).The joint PMF of any finite number of samples is

For example the probability of the first 10 samples being 1,0,1,0,1,0,1,0,1,0 is p5(1-p)5.We are able to specify the joint PMF for any finite number of sample times that is referred as being able to specify the finite dimensional distribution (FDD) .If the FDD does not change with the time origin

Such processes called stationary.

IID random process is stationaryTo prove that the IDD RP is a special case of a stationary RP we must show that the following equality holds

This follows from

By independenceBy identically distributedBy independenceIf a RP is stationary, then all its joint moments and more generally all expected values of the RP, must be stationary since

Non-stationary processesRP that are not stationary are ones whose means and/or variances change in time, which implies that the marginal PMF/PDF change with time.

mean increasing with nVariance decreasing with n

Sum random processSimilar to Random walk we have

The difference is that U[i] can have any, although the same PMF.Thus, the sum random process is not stationary since mean and variance change with n.

It is possible sometimes o transform a nonstationary RP into a stationary one.

Transformation of nonstationary RP into stationary one Example: for the sum RP this can be done by reversing the sum.

The difference or increment RV U[n] are IID. More generally

Nonoverlapping increments for a sum RP are independentIf furthermore, n4 n3 = n2 n1, then increments have same PMF since they are composed of the same number of IID RV.Such the sum processes, is said to have stationary independent increments (Random walk is one of them).

andBinomial counting random processConsider the repeated coin tossing experiment. We are interested in the number of heads that occur.Let U[n] be a Bernoulli random process

The number of heads is given by the binomial counting (sum)

or

The RP has stationary and independent increments.

Binomial counting random processLets determine pX[1],X[2][1,2] = P[X[1] = 1, X[2] =2].Note that the event X[1] = 1, X[2] = 2 is equivalent to the event Y1 = X[1] X[-1] = 1, Y2 = X[2] X[1] = 1, where X[-1] is defined to be identically zero.Y1 and Y2 are nonoverlapping increments (but of unequal length), making them independent RV, Thus

Example: Randomly phased sinusoidConsider the DTCV RP given as

where

= 3.43 = 6.01

Matlab codeThis RP is frequently used to model an analog sinusoid whose phase is unknown and that has been sampled by analog to digital convertor. Once two successive are observed, all the remaining ones are known.continuous17Joint momentsThe first (mean), second (variance) moments and covariance between two samples can always be estimated in practice, in contrast to the joint PMF, which may be difficult to determine.The mean and the variance sequence is defined as

The covariance sequence is defined as

Note that usual symmetry property of the covariance holds

whereand

Example: Randomly phased sinusoidRecalling that the phase is uniformly distributed ~ (0, 2) we have

21/2p()

For all n.19Example: Randomly phased sinusoidNoting that the mean sequence is zero, the covariance sequence becomes

The covariance sequence depends only on the spacing between the two samples or on n2 n1.Example: Randomly phased sinusoidNote the symmetry of the covariance sequence about n = 0.The variance follows as for all n.

Real-world Example Statistical Data AnalysisEarly we discussed an increase in the annual summer rainfall totals.Why questions is whether it supports global warming or not ?Lets fine exact increase of the rainfall by fitting a line an + b the historical data.

aReal-world Example Statistical Data AnalysisWe estimate a by fitting a straight line to the data set using a least squares procedure that minimizes the least square error (LSE)

To find b and a we perform

This results in two simultaneous linear equations

Where N = 108 four our data set.We used similar approach then were predicting a random variable outcome.

Real-world Example Statistical Data AnalysisIn vector/matrix form this is

Solving it we get estimation for a and b

Note that the mean indeed appears to be increasing with time.The LSE sequence is definedas The error can be quite large.

and

Real-world Example Statistical Data AnalysisThe increase is a = 0.0173 per year for a total increase of about 1.85 inches over the course of 108 years.Is it possible that the true value of a being zero?Lets assume that a is zero and then generate 20 realizations assuming the true mode is

Where U[n] is uniformly distributed process with var(U) = 10.05.

The estimating estimating a and b for each realization we get some of the estimated values of a are even negative.Real-world Example Statistical Data Analysis

Matlab codeHomeworkDescribe a random process that you are likely to encounter In the following situations.Listening to the daily weather forecastPaying the monthly telephone billLeaving for work in the morningWhy is each process random one?For a Bernoulli RP determine the probability that we will observe an alternating sequence of 1s and 0s for the first 100 samples with the first sample a 1. What is the probability that we will observe an alternating sequence of 1s and os for all n?Classify the following random processes as either Discrete-time/discrete valued, discrete-time/continuous valued, continuous valued/discrete value and continuous time/continuous value:Temperature in Rhode IslandOutcomes for continued spins of a roulette wheelDaily weight of personNumber of cars stopped at an intersection

HomeworkA random process X[n] is stationary. If it know that E[X[10]] = 10 and var(X[10]) = 1, then determine E[X[100]] and var(X[100]).A Bernoulli random process X[n] that takes on values 0 or 1, each with probability of p = , is transformed using Y[n] = (-1)nX[n]. Is the random process Y[n] IID?For the randomly phased sinusoid(see slide 19) determine the minimum mean square estimate of X[10] based on observing x[0]. How accurate do you think this prediction will be?For a random process X[n] the mean sequence X[n] and covariance sequence cX[n1,n2] are known. It is desired to predict k samples into the future. If x[n0] is observed, find the minimum mean square estimate of X[n0 + k]. Next assume that X[n] = cos(2f0n) and cX[n1, n2] = 0.9|n2 n1| and evaluate the estimate. Finally, what happens to your prediction as k and why?

HomeworkVerify that by differentiating with respect to b, setting the derivative equal to zero, and solving for b, we obtain the sample mean.

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