SEM 5

Random Signal Analysis

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Ques10 Bank

Syllabus Module No.

Content

Page nos.

1

Overview of Probability Theory and Basics of Random Variables 1.1 Sample Space, events, set operations, the notion and axioms of probability. 1.2 Conditional probability, Joint probabilty, Baye's rule, Independence of events, Sequential Experiments 1.3 Notion of random variable 1.4 Continuous random variables, probability density function, probability distribution function, Uniform, Exponential and Gaussain continuous random variables and distributions. 1.5 Discrete random variables, probability mass function, probability distribution function, binomial, Poisson and geometric discrete random variables and distributions.

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Operations on One Random Variable 2.1 Functions of a random variable and their distribution and density functions. 2.2 Expectations, Variance and Moments of random variable 2.3 Transformation of a random variable, Markov, Chebyshev and Chernoff bounds, characteristic functions, moment theorem.

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Multiple of Random Variables and Convergence 3.1 Vector random variables, Pairs of random variables, Joint CDF, Joint PDF, Independence, Conditional CDF and PDF, Conditional Expectation. 3.2 One function of two random variable, two functions of two random variables: joint moments, joint characteristic function, covariance and correlation-independent, uncorrelated and orthogonal random variables.

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Sequence of Random Variables and Convergence 4.1 Random sequences, Limit theorems: Strong and weak laws of larger numbers.

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Ques10 Bank 4.2 Central Limit Theorem and its significance 5

Random Process 5.1 Random process: Definition, realizations, sample paths, discrete and continuous time processes 5.2 Probabilistic structure of a Random process; mean, correlation and covariance functions, stationarity of random process. 5.3Ergodicity, Transmission of WSS random process through LTI system 5.4 Spectral analysis of random processes, power density spectrum bandwidth, cross power density spectrum 5.5 Gaussian and Poisson random process

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Markov Chains and Introduction to Queuing Theory 6.1 Markov processes 6.2 Discrete Markov chains, The n-step transition probabilities, steady state probabilites 6.3 Introduction to continuous time Markov chains 6.4 Classifications of states 6.5 Markovian models 6.6 Birth and death queuing models 6.7 Steady state results 6.8 Single and Multiple server Queuing Models 6.9 Finite source models 6.10 Little's Formula

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Distribution Matrix Module

Dec 2014

May 2015

Dec 2015

May 2016

1. Overview of Probability Theory and Basics of Random Variables 2. Operations on One Random Variable 3. Multiple of Random Variables and Convergence 4. Sequence of Random Variables and Convergence 5. Random Process

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40

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6. Markov Chains and Introduction to Queuing Theory

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35

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Overall Weightage

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Chap 1| Overview of Probability Theory and Basics of Random Variables Q1) State and prove Total Probability theorem and Bayes's theorem? Ans: [5M, 4M|May 2015, Dec 2014, Dec 2015, May 2016] Total Probability Theorem: Statement: If 𝐡1 , 𝐡2 , ……….𝐡𝑛 be a set of exhaustive and mutually exclusive events and A is another event associated with (or caused by) 𝐡𝑖 , then 𝑛

𝐴 𝑃(𝐴) = βˆ‘ 𝑃(𝐡𝑖 ). 𝑃 ( ) 𝐡𝑖 𝑖=1

Proof: The inner circle represents the event A. A can occur along with (or due to) 𝐡1 , 𝐡2 , ……….𝐡𝑛 that are exhaustive and mutually exclusive.

∴ 𝐴𝐡1 , 𝐴𝐡2 , 𝐴𝐡3 , 𝐴𝐡4 … … … … … … … . . 𝐴𝐡𝑛 are also mutually exclusive ∴ 𝐴 = 𝐴𝐡1 + 𝐴𝐡2 + 𝐴𝐡3 + 𝐴𝐡4 … … + 𝐴𝐡𝑛 (By Addition Theorem) 𝑛

∴ 𝑃(𝐴) = 𝑃 (βˆ‘ 𝐴𝐡𝑖 ) 𝑛

𝑖=1

= βˆ‘ 𝑃(𝐴𝐡𝑖 ) 𝑨

∴ 𝑷(𝑨) = βˆ‘π’π’Š=𝟏 𝑷(π‘©π’Š ). 𝑷 ( )…… probability

π‘©π’Š

𝑖=1

(A)

(Using conditional 𝑃(𝐴𝐡) = 𝑃(𝐴 ∩ 𝐡) =

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Ques10 Bank 𝐡

𝐴

𝐴

𝐡

𝑃(𝐡). 𝑃 ( ) = 𝑃(𝐴). 𝑃 ( )) Bayes’ Theorem or Theorem of Probability of causes Statement: If 𝐡1 , 𝐡2 , ……….𝐡𝑛 be a set of exhaustive and mutually exclusive events associated with a random experiment and A is another event associated with (or caused by) 𝐡𝑖 , then 𝐴 𝐡𝑖

𝑃(𝐡𝑖 ).𝑃( )

𝐡𝑖

𝑃( ) = 𝐴

i=1, 2 ….n

𝐴 𝐡𝑖

βˆ‘π‘› 𝑖=1 𝑃(𝐡𝑖 ).𝑃( )

Proof:

We know Conditional Probability is given as: 𝐴 𝐡 𝑃(𝐴𝐡𝑖 ) = 𝑃(𝐴 ∩ 𝐡𝑖 ) = 𝑃(𝐡𝑖 ). 𝑃 ( ) = 𝑃(𝐴). 𝑃 ( 𝑖 ) 𝐡𝑖

𝑃(𝐡𝑖 ).𝑃( )

𝐴

𝑃(𝐴)

Now using Total Probability Theorem we have 𝐴 𝑃(𝐴) = βˆ‘π‘›π‘–=1 𝑃(𝐡𝑖 ). 𝑃 ( ) 𝐡𝑖

(2)

(3)

𝑨 π‘©π’Š

π‘©π’Š

𝑷(π‘©π’Š ).𝑷( )

𝑨

βˆ‘π’ π’Š=𝟏 𝑷(π‘©π’Š ).𝑷( )

𝑷( ) =

𝐴 𝐡𝑖

𝐡

∴ 𝑃 ( 𝑖) =

From equation (2) and equation (3)

(1)

𝐴

𝑨 π‘©π’Š

……hence proved

Q2) A certain test for a particular cancer is known to be 95% accurate. A person submits to the test and the results are positive. Suppose that the person comes from a population of 100,000 where 2000 people suffer from that disease. What can we conclude about the probability that the person under test has that particular cancer? Ans: (5M| May 2015) Total Population n(S) =100000 Let β€˜A’ be the event that the person is under test, β€˜B1’ be the event that the person has cancer ∴ Probability that a person has the cancer is 6

Ques10 Bank 2000 = 0.02 100000 Hence probability that a person does not have the cancer: 𝑃(𝐡2 ) = 1 βˆ’ 𝑃(𝐡1 ) = 1 βˆ’ 0.02 = 0.98 Probability the test is positive when a person has cancer is: 𝑃(𝐡1 ) =

𝐴 95 )= = 0.95 𝐡1 100 Probability the test is positive when a person does not have the cancer is: 𝑃(

𝐴 ) = 1 βˆ’ 0.95 = 0.05 𝐡2 According to total probability theorem 𝒏 𝑨 𝑷(𝑨) = βˆ‘ 𝑷(π‘©π’Š ). 𝑷 ( ) π‘©π’Š π’Š=𝟏 𝑨 𝑨 𝑷(𝑨) = 𝑷 ( ) . 𝑷(π‘©πŸ ) + 𝑷 ( ) . 𝑷(π‘©πŸ ) π‘©πŸ π‘©πŸ 𝑷(𝑨) = 𝟎. πŸ—πŸ“ Γ— 𝟎. 𝟎𝟐 + 𝟎. πŸŽπŸ“ Γ— 𝟎. πŸ—πŸ– 𝑷(𝑨) = 𝟎. πŸŽπŸ”πŸ•πŸ Using Bayes Theorem 𝑃(

𝑨 π‘©π’Š

𝑩

𝑷(π‘©π’Š ).𝑷( )

𝑨

𝑷(𝑨)

𝑷 ( π’Š) =

𝐴 𝑃(𝐡1 ). 𝑃 ( ) 𝐡1 𝐡1 𝑃( ) = 𝐴 𝐴 𝑛 βˆ‘π‘–=1 𝑃(𝐡𝑖 ). 𝑃 ( ) 𝐡𝑖 π‘©πŸ 𝟎. 𝟎𝟐 Γ— 𝟎. πŸ—πŸ“ 𝑷( ) = = 𝟎. πŸπŸ–πŸπŸ• 𝑨 𝟎. πŸŽπŸ”πŸ•πŸ

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RSA-Ques10Bank-Sample.pdf

The response to KT280 Solutions was overwhelming. (Thank you guys!) So keeping to our promise. This time around, we have expanded six folds. serving Comps, IT, Extc, Electro, Mech, Civil for Sem1, Sem3, Sem5 and Sem7. Here's a sweet surprise! KT280 now provides you an open platform to share your answer writing.

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