Probability Random Variables And Random Processes Part 1

Probability, Random Variables and Random Processes with applications to Signal Processing Ganesh.G Member(Research Staff), Central Research Laboratory, Bharat Electronics Limited, Bangalore-13

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Notes :

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Organization of the Topic Probability, Random Variables and Random Processes with applications to Signal Processing

Probability

Part-1 02/May/2007

Random Variables with Applications to Signal Processing

Random Random Processes Processes with with Applications Applications to to Signal Signal Processing Processing

Part-2

Part-3 2 of 30

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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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Contents – 2. Random Variables • 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

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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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Contents – 2. Random Variables Contd.. • 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 02/May/2007

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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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Contents – 3. Random Processes • 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

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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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Contents – 3. Random Processes Contd.. • 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

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Notes :

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 02/May/2007

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Notes :

1. {R.G.Brown},pp1 2. {S&W},pp2 3. {S&W},pp2 4. {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 measure of Frequency of Occurrence

P[A] =

nE n

n

P[A] = Lim E n→∞ n

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Notes :

1. Refer their failures from {Papoulis} pp6-7 2. {S&W} pp2-4 3. 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 ( A) ≥ 0 (Probabili ty is a nonnegativ e number)

(i)

(ii) P (Ω ) = 1 (Probabili ty of the whole set is unity) (iii) If A ∩ B = φ , then P ( A ∪ B ) = P ( A) + P ( B ). - A.N.Kolmogorov • P(A1+ A2+

+ An) = 1

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1. Experiment, Sample Space, Elementary Event (Outcome), Event, Discuss the equations why they are so? - :Refer {Peebles},pp10 2. Axiomatic Theory Uses- Refer {Kolmogorov} 3. Consider a simple resistor R = V(t) / I(t) is this true under all conditions? Fully accurate?(inductance and capacitance?)clearly specified terminals? Refer{Papoulis}, 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

Before the Experiment is conducted

• A posteriori Probability – Reasoning from the observed facts

After the Experiment is conducted

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1. {S&W}, pp3 2. 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+ B1+ B2+

+ An be a partition of A and + Bn be a partition of B

• This leads to the Array of Joint and Marginal Probabilities

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Notes :

1. {R.G.Brown} pp12-13 2. Conditional probability, in contrast, usually is explained through relative frequency interpretation of probability see for example {S&W} pp16

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Concepts of Joint, Marginal, Conditional and Total Probabilities Marginal Probabiliti es

B

Event B1 Event B2

Event Bn

Event A1

P ( A1 ∩ B1 ) P( A1 ∩ B2 )

P( A1 ∩ Bn ) P(A1)

Event A2

P ( A2 ∩ B1 ) P ( A2 ∩ B2 )

P ( A2 ∩ Bn ) P(A2)

Event An

P ( An ∩ B1 )

P ( An ∩ Bn )

Marginal Probabilities

P(B1)

A

P(B2)

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P(Bn)

P(A2) SUM = 1

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Notes :

1. From {R.G.Brown} pp12-13 2. 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 Marginal Probabiliti es

B

Event B1 Event B2

Event Bn

Event A1

P ( A1 ∩ B1 ) P( A1 ∩ B2 )

P( A1 ∩ Bn ) P(A1)

Event A2

P ( A2 ∩ B1 ) P ( A2 ∩ B2 )

P ( A2 ∩ Bn ) P(A2)

Event An

P ( An ∩ B1 )

P ( An ∩ Bn )

Marginal Probabilities

P(B1)

A

P(B2)

P(Bn)

P(A2) SUM = 1

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Notes :

1. From {R.G.Brown} pp12-13 2. 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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Concepts of Joint, Marginal, Conditional and Total Probabilities • 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

P( A | B) = provided

P ( AB ) , P( B)

P( B) ≠ 0.

The above definition satisfies all probability axioms discussed earlier.

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Concepts of Joint, Marginal, Conditional and Total Probabilities • Let A1, A2, space A

An be a partition on the probability

• 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

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Notes :

1. {S&W} pp20 2. Average because expression looks likes averaging; Total because P(B) is sum of parts 3. In shade is ‘Total Probability Theorem’

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Baye’s Theorem and its applications • Bayes theorem:

– One form:

P ( AB ) , P( B)

P( A | B) =

hence,

P( A | B) =

P ( B | A) =

P ( AB ) P ( A)

P ( B | A) P ( A ) P(B)

– Other form:

P ( Ai | B ) =

P ( B | Ai ) P ( Ai ) n

∑

i =1

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,

P ( B | Ai ) P ( Ai )

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Notes :

1. {Peebles} pp16 2. 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

BSC

Transmit ‘0’ or ‘1’

P(0t) = 0.4 & Channel Effect P(1t) = 0.6

P(1r|1t) = 0.9 &

Receive ‘0’ or ‘1’ P(0r) = ? & P(1r) = ?

P(0r|1t) = 0.1 Symmetric; 0t is no different System Errors (BER)? Out of 100 Zeros/Ones received, how many are in errors? 02/May/2007

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Notes :

1. {Peebles} pp17 2. BSC Transition Probabilities

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Baye’s Theorem and its applications • 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

Out of 100 Zeros received, 14 are in errors

– P(0t|1r) = 0.069

Out of 100 Ones received, 6.9 are in errors

– P(0t|0r) = 0.931 02/May/2007

P(0t) = 0.4 & P(1t) = 0.6 20 of 30

Notes :

1. {Peebles} 2. 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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Independent events and their properties • Let two events A and B have nonzero probabilities of occurrence; assume P( A) ≠ 0 & P( B) ≠ 0. • 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,

P ( AB ) = P ( A ) ⋅ P ( B )

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Test for Independece 21 of 30

Notes :

1. {Peebles} pp19 2. 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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Independent events and their properties • 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:

P ( AB ) = P ( A ) ⋅ P ( B )

P ( BC ) = P ( B ) ⋅ P ( C ) P ( ABC ) = P ( A ) ⋅ P ( B ) ⋅ P ( C ) P ( AC ) = P ( A ) ⋅ P ( C ) 02/May/2007

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Notes :

1. {Peebles} pp19-20 2. 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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Independent events and their properties • 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.”

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Notes :

1. {Peebles} pp20

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Tips and Tricks • Single most difficult step in solving probability problems: Correct Mathematical Modeling

• 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 02/May/2007

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Notes :

1. {Peebles} and {Papoulis}

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Example “simple textbook examples” to practical problems of interest •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

− say m = np ; p < 1 −

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.

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Notes :

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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Example •Let “X = ( k + 1) stdrawn ball has the largest number among all n k balls, and the largest among the first k balls is in the group of first m balls, k > m.”

•Note that X k is of the form A ∩ B, where A = “largest among the first k balls is in the group of first m balls drawn”

and

B = “(k+1)st ball has the largest number among all n balls”. 02/May/2007

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Notes :

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

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Example Notice that A and B are independent events, and hence 1 m 1 np p P ( X k ) = P ( A) P ( B ) =

nk

=

n k

=

k

.

Where m = np represents the fraction of balls to be initially drawn. This gives P (“selected ball has the largest number among all balls”) = n −1 P( X ) = p n −1 1 ≈ p n 1 = p ln k n

∑

k =m

k

∑k

k =m

∫ np k

np

= − p ln p. 02/May/2007

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Example Maximization of the desired probability with respect to d p gives (− p ln p ) = −(1 + ln p ) = 0 dp or p = e−1 0.3679. The maximum value for the desired probability of drawing the largest number also equals 0.3679

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Notes : 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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What Next? •

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

Part - 2

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

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Part - 3

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References • 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, Univariate Discrete 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}

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Notes :

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

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