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Probability, Random Variables and Random Processes with applications to Signal Processing

Ganesh.G

Member(Research Staff),

Central Research Laboratory, Bharat Electronics Limited, Bangalore-13

ganesh.crl@gmail.com

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Random Processes with Applications to

Signal Processing

Probability

Random Variables with Applications to

Signal Processing

Probability, Random Variables and Random Processes with applications to Signal Processing

Random Variables with Applications to

Signal Processing

Random Processes with Applications to

Signal Processing

Probability

Part-1 Part-2 Part-3

Organization of the Topic

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Contents – 1. Probability

• Probability– Why study Probability– Four approaches to Probability definition– A priori and A posteriori Probabilities– Concepts of Joint, Marginal, Conditional and Total Probabilities– Baye’s Theorem and its applications– Independent events and their properties

• Tips and Tricks

• Example

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• The Concept of a Random Variable– Distribution and Density functions– Discrete, Continuous and Mixed Random variables– Specific Random variables: Discrete and Continuous– Conditional and Joint Distribution and Density functions

• Functions of One Random Variable– Transformations of Continuous and Discrete Random variables– Expectation– Moments: Moments about the origin, Central Moments, Variance

and Skew– Characteristic Function and Moment Generating Functions– Chebyshev and Shwarz Inequalities– Chernoff Bound

Contents – 2. Random Variables

1. Distribution and Density functions/ Discrete, Continuous and Mixed Random variables/ Specific Random variables: Discrete and Continuous/ Conditional and Joint Distribution and Density functions

2. Why functions of Random variables are important to signal processing/ Transformations of Continuous and Discrete Random variables/ Expectation/ Moments: Moments about the origin, Central Moments, Variance and Skew/ Functions that give Moments: Characteristic Function and Moment Generating Functions/ Chebyshev and Shwarz Inequalities/ Chernoff Bound

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• Multiple Random Variables– Joint distribution and density functions– Joint Moments (Covariance, Correlation Coefficient, Orthogonality)

and Joint Characteristic Functions– Conditional distribution and density functions

• Random Vectors and Parameter Estimation– Expectation Vectors and Covariance Matrices– MMSE Estimator and ML Estimator

• Sequences of Random Variables– Random Sequences and Linear Systems– WSS and Markov Random Sequences – Stochastic Convergence and Limit Theorems /Central Limit Theorem– Laws of Large Numbers

Contents – 2. Random Variables Contd..

1. Functions of two Random variables/ Joint distribution and density functions/ Joint Moments (Covariance, Correlation Coefficient, Orthogonality) and Joint Characteristic Functions/ Conditional distribution and density functions

2. Expectation Vectors and Covariance Matrices/ Linear Estimator, MMSE Estimator/ ML Estimators S&W*

3. Random Sequences and Linear Systems/ WSS Random Sequences /Markov Random Sequences S&W / Stochastic Convergence and Limit Theorems/ Central Limit Theorem Papoulis S&W/ Laws of Large Numbers S&W

*Note: Shown inside the brackets .. are codes for Reference Books. See page 30 of 30 of this document for references.

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• Introduction to Random Processes– The Random Process Concept– Stationarity, Time Averages and Ergodicity– Some important Random Processes

• Wiener and Markov Processes

• Spectral Characteristics of Random Processes

• Linear Systems with Random Inputs– White Noise– Bandpass, Bandlimited and Narrowband Processes

• Optimum Linear Systems– Systems that maximize SNR– Systems that minimize MSE

Contents – 3. Random Processes

1. Correlation functions of Random Processes and their properties –Peebles; S&W; Papoulis

2. Power Spectral Density and its properties, relationship with autocorrelation ; White and Colored Noise concepts and definitions-Peebles

3. Spectral Characteristics of LTI System response; Noise Bandwidth-Peebles;S&W

4. Matched Filter for Colored Noise/White Noise; Wiener Filters-Peebles

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• Some Practical Applications of the Theory– Noise in an FM Comm.System– Noise in a Phase-Locked Loop– Radar Detection using a single Observation– False Alarm Probability and Threshold in GPS

• Applications to Statistical Signal Processing– Wiener Filters for Random Sequences– Expectation-Maximization Algorithm(E-M)– Hidden Markov Models (and their specifications)– Spectral Estimation– Simulated Annealing

Contents – 3. Random Processes Contd..

1. Peebles; Consider ‘Code Acquisition’ scenario in GPS applications for one example in finding the false alarm rate

2. Kalman Filtering; Applications of HMM (Hidden Markov Model)s to Speech Processing –S&W

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Probability ………….Part 1 of 3

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Why study Probability

• Probability plays a key role in the description of noise like signals

• Nearly uncountable number of situations where we cannot make any categorical deterministic assertion regarding a phenomenon because we cannot measure all the contributing elements

• Probability is a mathematical model to help us study physical systems in an average sense

• Probability deals with averages of mass phenomena occurring sequentially or simultaneously:

– Noise, Radar Detection, System Failure, etc

1. R.G.Brown,pp12. S&W,pp23. S&W,pp24. Papoulis, pp1 [ 4.1Add Electron Emission, telephone calls,

queueing theory, quality control, etc.]5. Extra: Peebles pp2: [How do we characterize random signals:

One:how to describe any one of a variety of a random phenomena–Contents shown in Random Variables is required; Two: how to bring time into the problem so as to create the random signal of interest--Contents shown in Random Processes is required] – ALL these CONCEPTS are based on PROBABILITY Theory.

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Four approaches to Probability definition

• Probability as Intuition

• Probability as the ratio of Favorable to Total Outcomes

• Probability as a measureof Frequency of Occurrence

P[A] = Limn n

nE

∞→

P[A] = nnE

1. Refer their failures from Papoulis pp6-72. S&W pp2-43. Slide not required!? Only of Historical Importance?4. Classical Theory or ratio of Favorable to Total Outcomes approach

cannot deal with outcomes that are not equally likely and it cannot handle uncountably infinite outcomes without ambiguity.

5. Problem with relative frequency approach is that we can never perform the experiment infinite number of times so we can only estimate P(A) from a finite number of trails.Despite this, this approach is essential in applying probability theory to the real world.

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Four approaches to Probability definition

• Probability Based on an Axiomatic Theory

• P(A1+ A2+ + An) = 1

- A.N.Kolmogorov

).()()( then, If (iii)unity) isset whole theofty (Probabili 1)( (ii)

number) enonnegativa isty (Probabili 0)( (i)

BPAPBAPBAP

AP

+=∪=∩=Ω≥

φ

1. Experiment, Sample Space, Elementary Event (Outcome), Event, Discuss the equations why they are so? - :Refer Peebles,pp10

2. Axiomatic Theory Uses- Refer Kolmogorov3. Consider a simple resistor R = V(t) / I(t) - is this true

under all conditions? Fully accurate?(inductance and capacitance?)clearly specified terminals? ReferPapoulis, pp5

4. Mutually Exclusive/Disjoint Events? [(refer point (iii) above) when P(AB) = 0]. When a set of Events is called Partition/Decomposition/Exhaustive (refer last point in the above slide); what is its use?(Ans: refer Tips and Tricks page of this document )

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A priori and A posteriori Probabilities

• A priori Probability – Relating to reasoning from self-evident propositions or

presupposed by experience

• A posteriori Probability– Reasoning from the observed facts

Before the Experimentis conducted

After the Experimentis conducted

1. S&W, pp32. Also called ‘Prior Probability’ and ‘Posterior Probability’3. Their role; Baye’s Theorem: Prior: Two types: Informative Prior and

Uninformative(Vague/diffuse) Prior; Refer Kemp,pp41-42

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Concepts of Joint, Marginal, Conditional and Total Probabilities

• Let A and B be two experiments– Either successively conducted OR simultaneously

conducted

• Let A1+ A2+ + An be a partition of A and

B1+ B2+ + Bn be a partition of B

• This leads to the Array of Joint and Marginal Probabilities

1. R.G.Brown pp12-132. Conditional probability, in contrast, usually is explained through

relative frequency interpretation of probability see for exampleS&W pp16

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Concepts of Joint, Marginal, Conditional and Total Probabilities

SUM = 1P(B1) P(B2) P(Bn)Marginal Probabilities

P(A1)

P(A2)

Event A1

Event A2

Event An

Marginal Probabilities

Event B1 Event B2 Event BnB

A

)( 11 BAP ∩

)( 12 BAP ∩

)( 21 BAP ∩

)( 22 BAP ∩

)( 1BAP n ∩ )( nn BAP ∩

)( 2 nBAP ∩

)( 1 nBAP ∩

P(A2)

1. From R.G.Brown pp12-132. Joint Probability?3. What happens if Events A1,A2,….An are not a partition but just

some disjoint/Mutually Exclusive Events?Similarly for Events Bs?4. Summing out a row for example gives the probability of an event A

of Experiment A irrespective of the oucomes of Experiment A 5. Why they are called marginal? (because they used to be written in

margins)6. Sums of the Shaded Rows and Columns..

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Concepts of Joint, Marginal, Conditional and Total Probabilities

SUM = 1P(B1) P(B2) P(Bn)Marginal Probabilities

P(A1)

P(A2)

Event A1

Event A2

Event An

Marginal Probabilities

Event B1 Event B2 Event BnB

A

)( 11 BAP ∩

)( 12 BAP ∩

)( 21 BAP ∩

)( 22 BAP ∩

)( 1BAP n ∩ )( nn BAP ∩

)( 2 nBAP ∩

)( 1 nBAP ∩

P(A2)

1. From R.G.Brown pp12-132. This table also contains information about the relative frequency of

occurrence of various events in one experiment given a particular event in the other experiment .

3. Look at the Column with Red Box outline.Since no other entries of the table involve B2, list of these entries gives the relative distribution of events A1,A2,…..An given B2 has occurred.

4. However, Probabilities shown in the Red Box are not Legitimate Probabilities!(Because their sum is not unity, it is P(B2) ). So, imagine renormalizing all the entries in the column by dividing by P(B2). The new set of numbers then is P(A1.B2)/P(B2), P(A2.B2)/P(B2) … P(An.B2)/P(B2) and their sum is unity. And the relative distribution corresponds to the relative frequency of occurrence events A1,A2,…..An given B2 has occurred.

5. This heuristic reasoning leads us to the formal definition of ‘Conditional Probability’.

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• Conditional Probability : a measure of “the event A given that B has already occurred”. We denote this conditional probability by

P(A|B) = Probability of “the event A given that B has occurred”.

We define

provided

The above definition satisfies all probability axioms discussed earlier.

Concepts of Joint, Marginal, Conditional and Total Probabilities

,)()()|(

BPABPBAP =

.0)( ≠BP

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• Let A1, A2, An be a partition on the probability space A

• Let B be any event defined over the same probability space.

Then,

P(B) = P(B|A1)P(A1)+P(B|A2)P(A2)+ +P(B|An)P(An)

P(B) is called the “average” or “Total” Probability

Concepts of Joint, Marginal, Conditional and Total Probabilities

1. S&W pp202. Average because expression looks likes averaging; Total because

P(B) is sum of parts3. In shade is ‘Total Probability Theorem’

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Baye’s Theorem and its applications

• Bayes theorem:– One form:

hence,

– Other form:

,)()()|(

BPABPBAP =

)()()|(

APABPABP =

)()()|()|(

BPAPABPBAP =

,)()|(

)()|()|(

1∑

=

= n

iii

iii

APABP

APABPBAP

1. Peebles pp162. What about P(A) and P(B); both should not be zero or only P(B)

should not be zero?3. Ai’s form partition of Sample Space A; B is any event on the same

space

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Baye’s Theorem and its applications

• Consider Elementary Binary Symmetric Channel

System Errors (BER)? Out of 100 Zeros/Ones received, how many are in errors?

B S CTransmit ‘0’ or ‘1’

Receive ‘0’ or ‘1’

P(0t) = 0.4 &

P(1t) = 0.6

P(0r) = ? &

P(1r) = ?

Channel Effect

P(1r|1t) = 0.9 &

P(0r|1t) = 0.1

Symmetric; 0t is no different

1. Peebles pp172. BSC Transition Probabilities

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• P(0r) = 0.42 and P(1r) = 0.58 using total probability theorem (for each or P(1r) = 1- P(0r) )

• Using Baye’s Theorem:

– P(0t|0r) = 0.857

– P(1t|0r) = 0.143

– P(0t|1r) = 0.069

– P(0t|0r) = 0.931

Out of 100 Zeros received,14 are in errors

Baye’s Theorem and its applications

Out of 100 Ones received,6.9 are in errorsP(0t) = 0.4 &

P(1t) = 0.6

1. Peebles2. Average BER of the system is [(14 x 60 % )+ (6.9 x 40%) ] =

11.16% > 10% Erroneous Channel effect. This is due to unequal probabilities of 0t and 1t.

3. What happens if 0t and 1t are equi-probable? P(1t|0r) = 10% = P(0t|1r); and average BER of the system is [(10 x 50 % )+ (10 x 50%) ] = 10% = Erroneous Channel effect

4. Add: Bayesian methods of inference involve the systematic formulation and use of Baye’s Theorem. These approaches are distinguished from other statistical approaches in that, prior to obtaining the data, the statistician formulates degrees of belief concerning the possible models that may give rise to the data. These degrees of belief are regarded as probabilities. Kemp pp41 “ Posterior odds are equal to the likelihood ratio times the prior odds.” [Note:Odds on A = P(A)/P(Ac); Ac= A compliment]

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• Let two events A and B have nonzero probabilities of occurrence; assume

• We call these independent if occurrence of one event is not affected by the other event

P(A|B) = P(A) and P(B|A) = P(A)

• Consequently,

Independent events and their properties

& 0)( ≠AP .0)( ≠BP

)()()( BPAPABP ⋅=Testfor Independece

1. Peebles pp192. Can two independent events be mutually exclusive? Never (see the

first point in the slide; when both P(A) and P(B) are non-zero, how can P(AB) be zero? ).

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• Independence of Multiple Events: independence by pairs (pair-wise) is not enough.

• E.g., in case of three events A, B, C; the are independent if and only if they are independent pair-wise and are also independent as a triple, satisfying the following four equations:

Independent events and their properties

)()()( BPAPABP ⋅=

)()()( CPBPBCP ⋅=

)()()( CPAPACP ⋅=)()()()( CPBPAPABCP ⋅⋅=

1. Peebles pp19-202. How many Equations are needed for ‘N’ Events to be independent?

2^n – 1 – n (add 1+n to nc2+…+ncn and find what it is and subtract the same from that)

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• Many properties of independent events can be summarized by

the statement:

“If N events are independent then any of them is independent of any event formed by unions, intersections and complements of the others events.”

Independent events and their properties

1. Peebles pp20

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• Single most difficult step in solving probability problems:

• Many difficult problems can be solved by ‘going the other way’ and by recursion principle

• Model independent events in solving• Use conditioning and partitioning to solve tough

problems

Tips and Tricks

Correct Mathematical Modeling

1. Peebles and Papoulis

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•Day-trading strategy : A box contains n randomly numbered balls (not 1 through n but arbitrary numbers including numbers greater than n).

•Suppose a fraction of those balls are initially

drawn one by one with replacement while noting the numbers on those balls.

•The drawing is allowed to continue until a ball is drawn with a number larger than the first m numbers.

Determine the fraction p to be initially drawn, so as to maximize the probability of drawing the largest among the n numbers using this strategy.

Example

say ; 1m np p− = < −

“simple textbook examples” to practical problems of interest

1. Example and all notes relating to this example are taken with humble gratitude in mind from S.Unnikrishnan Pillai’s Web support for the book “A. Papoulis, S.Unnikrishnan Pillai, Probability, Random Variables and Stochastic Processes, 4th Ed: McGraw Hill, 2002”

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•Let “ drawn ball has the largest number among all n balls, and the largest among the first k balls is in the group of first m balls, k > m.”

•Note that is of the form where A = “largest among the first k balls is in the

group of first m balls drawn” andB = “(k+1)st ball has the largest number among

all n balls”.

Example

stk kX )1( +=

kX ,A B∩

1. P(A) = m/k and P(B) = 1/n

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Notice that A and B are independent events, and hence

Where m = np represents the fraction of balls to be initially drawn. This gives P (“selected ball has the largest number among all balls”)

Example

1 1

1 1( ) ln

ln .

n n n nk npnp

k m k mP X p p p k

k kp p

− −

= == = ≈ =

= −

∑ ∑ ∫

. 1 1)()()(kp

knp

nkm

nBPAPXP k ====

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Maximization of the desired probability with respect to p gives

or

The maximum value for the desired probability of drawing the largest number also equals 0.3679

Example

0)ln1()ln( =+−=− pppdpd

1 0.3679.p e−=

1. Interestingly the above strategy can be used to “play the stock market”. 2. Suppose one gets into the market and decides to stay up to 100 days. The stock

values fluctuate day by day, and the important question is when to get out?3. According to the above strategy, one should get out at the first opportunity after 37

days, when the stock value exceeds the maximum among the first 37 days. In that case the probability of hitting the top value over 100 days for the stock is also about 37%. Of course, the above argument assumes that the stock values over the period of interest are randomly fluctuating without exhibiting any other trend.

4. Interestingly, such is the case if we consider shorter time frames such as inter-day trading. In summary if one must day-trade, then a possible strategy might be to get in at 9.30 AM, and get out any time after 12 noon (9.30 AM + 0.3679 6.5 hrs = 11.54 AM to be precise) at the first peak that exceeds the peak value between 9.30 AM and 12 noon. In that case chances are about 37% that one hits the absolute top value for that day! (disclaimer : Trade at your own risk)

5. Author’s note: The same example can be found in many ways in other contexts, e.g., Puzzle No.34 “The Game of Googol” from M.Gardner; the ancient Indian concept of ‘Swayamvara’ to name a few.

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• Random Variables with applications to Signal Processing– The Concept of a Random Variable– Functions of One Random Variable– Multiple Random Variables– Random Vectors and Parameter Estimation– Sequences of Random Variables

• Random Processes with Applications to Signal Processing– Introduction to Random Processes– Spectral Characteristics of Random Processes– Linear Systems with Random Inputs– Optimum Linear Systems– Some Practical Applications of the Theory– Applications to Statistical Signal Processing

What Next?

Part - 2

Part - 3

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• 1. A. Papoulis, S.Unnikrishnan Pillai, Probability, Random Variables and Stochastic Processes, 4th Ed: McGraw Hill,2002. Papoulis

• 2. Henry Stark, John W.Woods, Probability and Random Processes with Applications to Signal Processing,3rd Ed: Pearson Education, 2002. S&W

• 3. Peebles Peyton Z., Jr, Probability, Random Variables and Random Signal Principles,2nd Ed: McGraw Hill,1987. Peebles

• 4. Norman L.Johnson, Adrienne W.Kemp, Samuel Kotz, UnivariateDiscrete Distributions, 3rd Ed: Wiley, 2005. Kemp

• 5. A.N.Kolmogorov, Foundations of the Theory of Probability: Chelsea, 1950. Kolmogorov

• 6. Robert Grover Brown, Introduction to Random Signal analysis and Kalman Filtering: John Wiley,1983. R.G.Brown

• 7. J.L.Doob, Stochastic Processes:John Wiley,1953 Doob• 8. Martin Gardner, My Best Mathematical and Logic Puzzles: Dover

Publications, Inc, New York, 1994. M.Gardner

References

1. Shown in the brackets are the codes used to annotate them in the notes area.