MA6451 PROBABILITY AND RANDOM PROCESSES - Mathematics... · UNIT III RANDOM PROCESSES 3.2 Stationary…

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<ul><li><p>MA6451 PROBABILITY AND RANDOM PROCESSES </p><p>UNIT I RANDOM VARIABLES </p><p>1.1 Discrete and continuous random variables </p><p>1. Show that the function is a probability density function of a </p><p>random variable X. (Apr/May 2015)3 </p><p>2. A continuous random variable X that can assume any value between X=2 and X=5 has a </p><p>probability density function given by f(x)=k(1+x). Find P(X </p></li><li><p>UNIT II TWO - DIMENSIONAL RANDOM VARIABLES </p><p>2.1 Joint distributions </p><p>1. Find the value of k, if f(x,y) = kxye-(x2+y2) : x0, y0 is to be a joint probability density function. </p><p>(Apr/ May 2015)3 </p><p>2.2 Marginal and conditional distributions </p><p>1. If the joint probability distribution function of a two dimensional random variable (X,Y) is </p><p>given by , find the marginal densities of X and Y. Are </p><p>X and Y independent? Find P[1</p></li><li><p>UNIT III RANDOM PROCESSES </p><p>3.2 Stationary process </p><p>1. Give an example of evolutionary random process.(Apr/May 2015)3 </p><p>2. Show that the process X(t) = A cos t + B sin t where A and B are random variables, is wide </p><p>sense stationary process if E(A) = E (B) = E(AB) = 0, E(A2) = E(B2).(Apr/May 2015)3 </p><p>3.4 Poisson process </p><p>1. A radioactive source emits particles at a rate of 5 per minute in accordance with Poisson </p><p>process. Each particle emitted has a probability 0.6 of being recorded. Find the probability </p><p>that 10 particles are recorded in 4 minute period.(Apr/May 2015)3 </p><p>3.5 Discrete parameter Markov chain </p><p>1. Define a semi-random telegraph signal process.(Apr/May 2015)3 </p><p>2. Check if a random telegraph signal process is wide sense stationary.(Apr/May 2015)3 </p><p>3.7 Limiting distributions. </p><p>1. There are 2 white marbles in Urn A and 3 red marbles in Urn B. At each step of the process, </p><p>a marble is selected from each urn and the 2 marbles selected are interchanged. The state </p><p>of the related Markov chain is the number of red marbles in Urn A after the interchange. </p><p>What is the probability that there are 2 red marbles in Urn A after 3 steps? In the long run, </p><p>what is the probability that there are 2 red marbles in Urn A?(Apr/May 2015)3 </p></li><li><p>UNIT IV CORRELATION AND SPECTRAL DENSITIES </p><p>4.1 Auto correlation functions </p><p>1. Find the auto correlation function whose spectral density is </p><p>(Apr/May 2015)3 </p><p>2. Consider two random processes X(t)=3 cos (t + ) and Y(t) = 2 cos (t + ), where </p><p>and is uniformly distributed over (0, 2). Verify (Apr </p><p>/May 2015)3 </p><p>4.2 Cross correlation functions Properties </p><p>1. State any two properties of cross correlation function.(Apr/May 2015)3 </p><p>4.3 Power spectral density </p><p>1. Find the Power spectral density of a random binary transmission process where </p><p>autocorrelation function is (Apr/May 2015)3 </p><p>2. If the power spectral density of a continuous process is , find the </p><p>mean square value of the process.(Apr/May 2015)3 </p><p>4.4 Cross spectral density Properties </p><p>1. A stationary process has an autocorrelation function given by . Find the </p><p>mean value, mean-square value and variance of the process.(Apr/May 2015) 3 </p></li><li><p>UNIT V LINEAR SYSTEMS WITH RANDOM INPUTS </p><p>5.1 Linear time invariant system- System transfer function </p><p>1. State the relation between input and output of a linear time invariant system.(Apr/May </p><p>2015)3 </p><p>2. If the input to a time invariant stable linear system is a wide sense stationary process, prove </p><p>that the output will also be a wide sense stationary process.(Apr/May 2015)3 </p><p>3. Show that where SXX() and SYY() are the power spectral </p><p>densities of the input X(t) and output respectively and H() is the system transfer </p><p>function.(Apr/May 2015)3 </p><p>5.2 Linear systems with random inputs </p><p>1. Prove that Y(t)=2X(t) is linear.(Apr/May 2015)3 </p><p>2. A circuit has an impulse response given by . Express SYY() in terms of </p><p>SXX().(Apr/May 2015)3 </p><p>5.3 Auto correlation and Cross correlation functions of input and output </p><p>1. Given and where Find the </p><p>spectral density of the output Y(t).(Apr/May 2015)3 </p></li></ul>

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