prof. sankarreview of random process1 probability sample space (s) –collection of all possible...
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Prof. Sankar Review of Random Process 1
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. Sankar Review of Random Process 2
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
NA- number of times event A occurred
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Prof. Sankar Review of Random Process 3
Probability
• Properties
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Prof. Sankar Review of Random Process 4
Probability
• Conditional Probability– Probability of B conditioned by the fact that A
has occurred
– The two events are statistically independent if
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Prof. Sankar Review of Random Process 5
Probability
• Bernoulli’s Trials– Same experiment repeated n times to find the
probability of a particular event occurring exactly k times
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Prof. Sankar Review of Random Process 6
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 nth trial
Prof. Sankar Review of Random Process 7
Random Signals• Mean
• Median: Middle or most central item in an
ordered set of numbers
• Mode = Max{xi}
• Variance
• Standard Deviation measure of spread or deviation from the mean
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Prof. Sankar Review of Random Process 8
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. Sankar Review of Random Process 9
Random Variables• Cumulative Distribution Function (CDF)
– Properties:
• Probability Density Function (PDF)
– Properties:
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Prof. Sankar Review of Random Process 10
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.
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Prof. Sankar Review of Random Process 11
Important Distributions
• Binomial Distribution– Perform binary experiment n times with
outcome X1,X2,…Xn, if , then X has binomial distribution
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Prof. Sankar Review of Random Process 12
Important Distributions
• Uniform Distribution– Random variable is equally likely– Equally Weighted pdf
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Prof. Sankar Review of Random Process 13
Important Distributions• Poisson Distribution
– Random Variable is Poisson distributed with parameter m with
– Approximation to binomial with p << 1,
and k << 1, then
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Prof. Sankar Review of Random Process 14
Important Distributions• Gaussian Distribution
• Normalized Gaussian pdf - N(0,1)– Zero mean, Unit Variance
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Prof. Sankar Review of Random Process 15
Important Distributions
• Normalized Gaussian pdf
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Prof. Sankar Review of Random Process 16
Joint and Conditional PDFs
• For two random variables X and Y–
(y)(x)pp(x,y)ptindependen are Yand X If
pdf under the Volume
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Prof. Sankar Review of Random Process 17
Joint and Conditional PDFs• Marginal pdfs
• Conditional pdfs
(x,y)dxp(y)p
(x,y)dyp(x)p
YXY
YXX
(x)p
(x,y)px)|(yp
(y)p
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Prof. Sankar Review of Random Process 18
Expectation and Moments
•
Centralized Moment
– Second centralized moment is variance
dx)x(px)x(E:X ofmoment n
)statisticordernd2(dx)x(px)x(E:X ofmoment Second
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Prof. Sankar Review of Random Process 19
Expectations and Moments
• (i,j) joint moment between random variables X and Y
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Prof. Sankar Review of Random Process 20
Expectations and Moments• (i,j) joint central moment
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Prof. Sankar Review of Random Process 21
Expectations and Moments• Auto-covariance
• Characteristic Function (moment generator)
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Prof. Sankar Review of Random Process 22
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. Sankar Review of Random Process 23
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. Sankar Review of Random Process 24
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)
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Prof. Sankar Review of Random Process 25
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
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Prof. Sankar Review of Random Process 26
Random Process
• Mean – Mean of the random process at time t is the mean of the
random variable X(t) • Autocorrelation
• Auto-covariance
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Prof. Sankar Review of Random Process 27
Random Process• Cross Correlation and covariance
• Power Density Spectrum
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Prof. Sankar Review of Random Process 28
Random Process
• Total Average Power
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