Question Paper Code : AEC11T03

Hall Ticket No Question Paper Code : AEC11T03 VARDHAMAN COLLEGE OF ENGINEERING (AUTONOMOUS) Affiliated to JNTUH, Hyderabad Four Year B Tech III Semester Tutorial Question Bank 2013-14 (Regulations: VCE-R11) PROBABILITY THEORY AND STOCHASTIC PROCESSES (Electronics and Communication Engineering) PART-A Unit-I 1 Define random Variable 2 The first four moments of a distribution about x=4 are 1,4,10 and 45 respectively Show that the mean is 5, variance is 3, μ 3 = 0 and μ 4 = 26 3 Define moment generating function and write the formula to find mean and variance 4 Find the moment generating function of binomial distribution 5 The mean and variance of the binomial distribution are 4 and 3 respectively Find P(X=0) 6 State any two instances where Poisson distribution may be successfully employed 7 In which probability distribution, variance and mean are equal 8 Write the moment generating function of Geometric distribution 9 Define generalized form of the gamma distribution 10 Write two characteristics of the Normal distribution Unit-II 1 Define joint distributions of two random variables X and Y and state its properties 2 If two random variables X and Y have pdf f(x,y) = k(2x+y) for 0 x 2, 0 y 3, evaluate k 3 If Y=-2x+3, find the cov(x, y) 4 Prove that the correlation coefficient P xy takes value in the range -1 to 1 5 Distinguish between correlation and regression 6 The regression equations of two variables x and y are 3x+y=10 and 3x+4y=12 find the coefficient of correlation between them 7 If x and y are independent random variable with variance 2 and 3 Find the variance of 3x+4y 8 State central limit theorem 9 State central limit theorem in Liapunoff s form 10 11 State central limit theorem in Lindberg-Levy s form Write the applications of central limit theorem Unit-III 1 State the four types of stochastic processes 2 {X(s,t)} is a random process, what is the nature of X(s,t) when (a) s is fixed (b) t is fixed? 3 Define strict sense stationary process 4 Define wide sense stationary process 5 State Chapman-kolmogorov theorem 6 When is a stochastic process said to be ergodic? 7 Give an example of an ergodic processes 8 Define Markov chain and one-step transition probability 9 Define Markov process 10 Define Binomial process 11 State the properties of Bernoulli process 12 Prove that the sum of two independent Poisson process is a Poisson process 13 State any two properties of Poisson process

Unit-IV 1 State any two properties of an auto correlation function 2 Define cross correlation and its properties 3 Prove that R XY (t) = R YX (-t) 4 State any two properties of cross correlation 5 Define Spectral density 6 What is meant by spectral analysis? 7 State any two uses of spectral density 8 Define cross spectral density and its examples 9 State Wiener Khintchine relation 10 State any two properties of cross-power density spectrum Unit-V 1 Define White Noise 2 Define thermal noise 3 Define Band Limited White Noise 4 Find the autocorrelation function of a Gaussian white noise 5 Define average noise bandwidth 6 Define effective noise temparature 7 Define average noise figure PART-B Unit-I 1 The take-off roll distance for aircraft at a certain airport can be any number from 80 m to 1750 m propeller aircraft require from 80 m to 1050 m while jets use from 950 m to 1750 m The overall runaway is 2000 m Determine sets A, B and C defined as propeller aircraft take-off distances and run away length safety margin respectively Determine the set A B and give its physical significance Determine the set and state the meaning of the set Determine the sets and and state the meanings of the sets and 2 A man wins in a gambling game if he gets two heads in in five flips of a biased coin The probability of getting a head with the coin is 07 Find the probability the man will win Should he play this game? What is the probability of winning if he wins by getting at least four heads in five flips? Should he play this new game? 3 A random variable X has the density function Define the events A= {1 < x 3} B={X 25} and C= A B Find the probabili es of the events 1) A 2) B and 3) C 4 In a certain Junior Olympics javelin throw distances are well approximated by a Gaussian distribution for which a X =30 m and σ X = 5 m In a qualifying round, contestants must throw farther than 26 m to qualify In the main event the record throw is 42 m What is the probability of being disqualified in the qualifying round? In the main event, what is the probability the record will be broken? 5 A certain large city averages three murders per week and their occurrences follows a Poisson distribution What is the probability that there will be five or more murders in a given week? On the average, how many weeks a year can this city expect to have no murders? How many weeks per year (average) can the city expect the number of murders per week to equal or exceed the average number per week? 6 In a game, show contestants choose one of three doors to determine what prize they win History shows that the three doors 1, 2 and 3, are chosen with probabilities 03, 045 and 025 respectively It is also known that given door 1 is chosen, the probabilities of winning prizes of $0, $100 and $1000 are 01, 02 and 07 For door 2 the respective probabilities are 05, 035 and 015 and for door 3 they are 08, 015 and 005 If X is a random variable describing dollars won, and D describes the door selected ( values of D are D 1 =1, D 2 =2 and D 3 =3), find

7 In the experiment of throwing two fair dice, let A be the event that the first die is odd, B be the event that the second die is odd, and C is the event that the sum is odd Show that events A, B and C are pair wise independent, but A, B and C are not independent 8 A random variable X has the following probability distribution x 0 1 2 3 4 5 6 7 9 P(x) 0 K 2K 2K 3K K 2 2K 2 7K 2 +K Find The value of K P(15 < x < 45 / X > 2) The smallest value of λ for which P(X λ) > ½ The arcsine probability density is defined by 10 11 12 13 For any real constants Show that for this density A random variable X can have values each with probability Find the density function, the mean and the variance of the random variable An experiment consists of rolling a single die Two events are defined as: A={a 6 shows up} and B={a 2 or a 5 shows up} 1) Find and 2) Define a third event C so that A company sells high fidelity amplifiers capable of generating 10, 25, and 50 W of audio power It has on hand 100 of the 10-W units, of which 15% are defective, 70 of the 25-W units with 10% defective, and 30 of the 50-W units with 10% defective What is the probability that an amplifier sold from the 10-W units is defective? If each wattage amplifier sells with equal likelihood, what is the probability of a randomly selected unit being 50W and defective? What is the probability that a unit randomly selected for sale is defective? A random variable X has the distribution function 14 15 16 17 18 19 20 Find the probabilities: 1) 2) 3) Write the density and distribution functions of binomial and Poisson random variables Show that the mean value and variance of the random variable having the uniform density function are and Define moments about the origin and central moments of the random variable X An experiment has a sample space with 10 equally likely elements Three events are defined as, and Find the probabilities of 1)A U C 2) B U C 3) A (B U C) 4) (A U B) C Two cards are drawn from a 52 card deck (the first is not replaced) Given the first card is a queen, what is the probability that the second is also a queen? Repeat part a) for the first card a queen and the second card a 7 What is the probability that both cards will be a queen? Spacecraft are expected to land in a prescribed recovery zone 80% of the time Over a period of time, six spacecrafts land Find the probability that none lands in the prescribed zone Find the probability that at least one will land in the prescribed zone The landing program is called successful if the probability is 09 or more that three or more out of six spacecraft will land in the prescribed zone Is the program successful? A man matches coin flips with a friend He wins 2 Rs if coins match and loses 2 Rs if they do not match Sketch a sample space showing possible outcomes for this experiment and illustrate how the points map onto the real line x that defines the values of the random variable X= dollars won on a trial Show a second mapping for a random variable Y= dollars won by the friend on a trial Unit-II

1 Discrete random variables X and Y have joint distribution function Find The marginal distributions and and sketch the two functions 2 Given the function Find the constant b such that this is a valid joint density function Determine the marginal density functions 3 Random variables X and Y have respective density functions Find and sketch the density function of if X and Y are independent 4 Two Gaussian random variables X and Y have a correlation coefficient The standard deviation of X is A linear transformation (coordinate rotation of ) is known to transform X and Y to new random variables that are statistically independent What is? 5 Two random variables X and Y have means =1 and =2, variances and and a correlation coefficient New random variables W and V are defined by Find the mean, variances, correlations and the correlation coefficient 6 Find a constant b (in terms of a) so that the function of and is a valid joint density function Find an expression for the joint distribution function Also find the marginal density functions 7 Let X and Y be statistically independent random variables with,,, and For a random variable, find,, and Are X and Y uncorrelated? 8 A joint probability density function is Find If, find 1) 2) 9 Two random variables X and Y have the density function Find all the first and second order moments Find the covariance Are X and Y uncorrelated? Unit-III 1 Given the random process Where A and are constants and is a random variable uniformly distributed on the interval (-π, π) Define a new random process Find the autocorrelation function of Find the cross-correlation function of X Are X wide-sense stationary? Are X jointly wide-sense stationary? 2 Given the random process Where A and are constants and is a random variable uniformly distributed on the interval (-π, π) Define a new random process Find the autocorrelation function of Find the cross-correlation function of X Are X wide-sense stationary? Are X jointly wide-sense stationary?

3 A Gaussian random process has an autocorrelation function Determine a covariance matrix for the random variables X(t), X(t+1), X(t+2), and X(t+3) 4 Let jointly wide-sense stationary processes and cause responses and respectively from a linear time-invariant system with impulse response h(t) If the sum is applied, the response is Find expressions, in terms of h(t) and characteristics of and, for a) b) 5 Discuss the mean and mean squared value of system response 6 Given the random process ) where A and are constants and is a random variable uniformly distributed on the interval (-π, π) Define a new random process Find the autocorrelation function of Find the cross correlation function of and Are and wide-sense stationary? Are and jointly wide-sense stationary? 7 A random process is defined by ) where is a wide-sense stationary random process that amplitude modulates a carrier of constant angular frequency with a random phase independent of and uniformly distributed on (-π, π) Find E[Y(t)] Find the autocorrelation function of Y(t) Is Y(t) wide-sense stationary? 8 Given the random process ) where is a constant, and A and B are uncorrelated zero-mean random variables having different density functions but the same variances Show that X(t) is wide-sense stationary but not strictly stationary 9 Statistically independent, zero-mean, random processes X(t) and Y(t) have autocorrelation functions and respectively Find the autocorrelation function of the sum Find the autocorrelation function of the difference Find the cross-correlation function of and 10 Given two random processes X(t) and Y(t), find the expressions for the auto-correlation function of if: are correlated They are uncorrelated They are uncorrelated with zero means Unit-IV 1 We are given the random process where A and are constants and is a random variable uniformly distributed on the interval (0, π) Is wide-sense stationary? Find the powers in 2 Work problem 1 if the process is defined by where is the unit step function 3 Work problem 2 assuming is a random variable uniformly distributed on the interval (0, π/2) 4 Work problem 1 if the random process is given by 5 Let A and B be random variables, we form the random process where is a real constant Show that if A and B are uncorrelated with zero means and equal variances, then is wide sense stationary Find the autocorrelation function of Find the power density spectrum 6 A random process is defined by where is a lowpass wide-sense stationary process, w 0 is a real constant and is a random variable uniformly distributed on the interval (0, 2π) Find and sketch the power density spectrum of in terms of that of Assume is independent of 7 Determine which of the following functions can and cannot be valid power density spectrums For those that are not, explain why a) b) c) d)

8 Determine which of the following functions can and cannot be valid power density spectrums For those that are not, explain why a) b) c) d) 9 If X(t) is a stationary process, find the power spectrum of in terms of the power spectrum of X(t) if A and B are real constants 10 A random process has the power density spectrum, find the average power in the process 11 A random process has the power density spectrum, find the average power in the process 12 A random process has the power density spectrum, find the average power in the process 13 A random process is given by where A and B are real constants, X(t) and Y(t) are jointly wide-sense stationary process a Find the power spectrum of w(t) b Find if X(t) and Y(t) are uncorrelated 14 Unit-V 1 Consider the white Gaussian Noise of zero mean and power spectral density No/2 applied to a low pass RC filter where transfer function is Find the output spectral density and auto correlation function of the output process 2 Find the input auto correlation function, output auto correlation function and output spectral density of the RC low pass filter when the filter is subjected to a white noise of spectral density 3 Define white noise Find the ACF of the white noise 4 If y(t) = A cos (wot +q) + N(t) where A is a constant, q is a random variable with a uniform distribution in (-p,p) and {N(t)} is a band limited Gaussian white noise with a power spectral density Find the power spectrum density of {Y(t)} Assume that N(t) and q are independent