Prof. SankarReview of Random Process1 Probability Sample Space (S) –Collection of all possible outcomes of a random experiment Sample Point –Each outcome of the experiment (or) element in the sample space Events are Collection of sample points Ex: Rolling a die (six sample points), Odd number thrown in a die (three sample point – a subset), tossing a coin (two sample points: head,tail)

Prof. SankarReview of Random Process2 Probability Null Event (No Sample Point) Union (of A and B) –Event which contains all points in A and B Intersection (of A and B) –Event that contains points common to A and B Law of Large Numbers – N – number of times the random experiment is repeated N A - number of times event A occurred

Prof. SankarReview of Random Process3 Probability Properties

Prof. SankarReview of Random Process4 Probability Conditional Probability –Probability of B conditioned by the fact that A has occurred –The two events are statistically independent if

Prof. SankarReview of Random Process5 Probability Bernoulli’s Trials –Same experiment repeated n times to find the probability of a particular event occurring exactly k times

Prof. SankarReview of Random Process6 Random Signals Associated with certain amount of uncertainty and unpredictability. Higher the uncertainty about a signal, higher the information content. –For example, temperature or rainfall in a city –thermal noise Information is quantified statistically (in terms of average (mean), variance, etc.) Generation –Toss a coin 6 times and count the number of heads –x(n) is the signal whose value is the number of heads on the n th trial

Prof. SankarReview of Random Process7 Random Signals Mean Median: Middle or most central item in an ordered set of numbers Mode = Max{x i } Variance Standard Deviation measure of spread or deviation from the mean

Prof. SankarReview of Random Process8 Random Variables Probability is a numerical measure of the outcome of the random experiment Random variable is a numerical description of the outcome of a random experiment, i.e., arbitrarily assigned real numbers to events or sample points –Can be discrete or continuous –For example: head is assigned +1 tail is assigned –1 or 0

Prof. SankarReview of Random Process9 Random Variables Cumulative Distribution Function (CDF) –Properties: Probability Density Function (PDF) –Properties:

Prof. SankarReview of Random Process10 Important Distributions Binary distribution (Bernoulli distribution) –Random variable has a binary distribution –Partitions the sample space into two distinct subsets A and B –All elements in A are mapped into one number say +1 and B to another number say 0.

Prof. SankarReview of Random Process11 Important Distributions Binomial Distribution –Perform binary experiment n times with outcome X 1,X 2,…X n, if, then X has binomial distribution

Prof. SankarReview of Random Process12 Important Distributions Uniform Distribution –Random variable is equally likely –Equally Weighted pdf ab

Prof. SankarReview of Random Process13 Important Distributions Poisson Distribution –Random Variable is Poisson distributed with parameter m with –Approximation to binomial with p << 1, and k << 1, then

Prof. SankarReview of Random Process14 Important Distributions Gaussian Distribution Normalized Gaussian pdf - N(0,1) –Zero mean, Unit Variance

Prof. SankarReview of Random Process15 Important Distributions Normalized Gaussian pdf

Prof. SankarReview of Random Process16 Joint and Conditional PDFs For two random variables X and Y –

Prof. SankarReview of Random Process17 Joint and Conditional PDFs Marginal pdfs Conditional pdfs

Prof. SankarReview of Random Process18 Expectation and Moments Centralized Moment –Second centralized moment is variance

Prof. SankarReview of Random Process19 Expectations and Moments (i,j) joint moment between random variables X and Y

Prof. SankarReview of Random Process20 Expectations and Moments (i,j) joint central moment

Prof. SankarReview of Random Process21 Expectations and Moments Auto-covariance Characteristic Function (moment generator)

Prof. SankarReview of Random Process22 Random Process If a random variable X is a function of another variable, say time t, x(t) is called random process Collection of all possible waveforms is called the ensemble Individual waveform is called a sample function Outcome of a random experiment is a sample function for random process instead of a single value in the case of random variable

Prof. SankarReview of Random Process23 Random Process Random Process X(.,.) is a function of time variable t and sample point variable s Each sample point (s) identifies a function of time X(.,s) referred as “sample function” Each time point (t) identifies a function of sample points X(t,.), i.e., a random variable Random or Stochastic Processes can be –continuous or discrete time process –continuous or discrete amplitude process

Prof. SankarReview of Random Process24 Random Process Ensemble statistic : Ensemble average at a particular time –Temporal average for a sample function Random Process Classifications –Stationary Process : Statistical characteristics of the sample function do not change with time (time-invariant)

Prof. SankarReview of Random Process25 Random Process Second Order joint pdf –Autocorrelation is a function of only time difference Wide Sense (or Weak) Stationary –Independent of time up to second order only Ergodic Process –Ensemble average = time average

Prof. SankarReview of Random Process26 Random Process Mean –Mean of the random process at time t is the mean of the random variable X(t) Autocorrelation Auto-covariance

Prof. SankarReview of Random Process27 Random Process Cross Correlation and covariance Power Density Spectrum

Prof. SankarReview of Random Process28 Random Process Total Average Power