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- Probability and Random Processes - PRP
- 2018
- PYQ
**Biju Patnaik University of Technology Rourkela Odisha - BPUT**- Electronics and Communication Engineering
- B.Tech
**82 Views**- 4 Offline Downloads
- Uploaded 10 months ago

Registration No : Total Number of Pages : 02 B.Tech PET3D001 3rd Semester Regular Examination 2018-19 PROBABILITY AND RANDOM PROCESSES BRANCH : ECE, ETC Time : 3 Hours Max Marks : 100 Q.CODE : E977 Answer Question No.1 (Part-1) which is compulsory, any eight from Part-II and any two from Part-III. The figures in the right hand margin indicate marks. Q1 a) b) c) d) e) f) g) h) i) j) Q2 a) b) c) d) e) f) g) h) i) Part- I Short Answer Type Questions (Answer All-10) (2 x 10) Define joint probability. Define conditional probability. Let X be a Binomial random variable with n=10 and p=0.6. Then E(X)=________ and Var (X)=_________ Two cards are drawn at random from a pack of 52 playing cards. Find the probability that both the cards are either kings or queens. Let X be a Poisson random variable with expectation 4. Then P(X=0)=________ Define Probability density function. Define Gaussian random variable. State the moment generating function of a random variable X about the origin. The expression for the distribution function of the Binomial random variable (with parameters n and p) using the unit step function is ______ If E(X2) = E (Y2) =5, E (XY) =3, V= 2X-Y, W = -3X+2Y then find out E (VW). State central limit theorem. Part- II Focused-Short Answer Type Questions- (Answer Any Eight out of Twelve) (6 x 8) Explain about the total probability. 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}. Find P (A) and P (B). Define a third event C so that P(C) = 1–P(A) –P(B) Two persons are competing for the post of principal of a college. The probability that the first and second person will win are 0.6 and 0.4 respectively. If the first person wins the probability of introducing the common model examination is 0.8 and the corresponding probability if the second person wins is 0.3.What is the probability that the common model examination will be introduced? What are the conditions for a function to be a random variable? Differentiate between discrete and continuous random variable. Write the methods of defining conditioning event and also derive conditional density function. A random variable X is a function. So is probability P. The domain of a function is the set of values its argument may take on while its range is the set of corresponding values of the function. In terms of sets, events, and sample spaces , stat the domain and range for X and P. The number of a car arriving at a certain drive- in window a bank is a Poisson random variable with expectation 2. Find the probability that more than three cars will arrive during a 10minutes period. Derive the moments about the origin and about the mean value of a random variable X? Explain about the monotonic transformation of a continuous random variable.

j) k) l) Q3 Q4 Q5 Q6 A discrete random variable X has possible values xi=i2, i=1, 2,3,4,5 which occur with probabilities 0.4, 0.25, 0.15, 0.1 and 0.1 respectively. Find the mean value X = E[X] of X. Find out the conditional density f (y|x) from the joint density ( ) f , (x, y) = u(x)u(y)xe Differentiate between the unequal and equal distributions of central limit theorem? Part-III Long Answer Type Questions (Answer Any Two out of Four) a) State and prove Bayes Rule. b) Box A contains 10 red and 7 blue marbles. Box B contains 8 red and 12 blue marbles. If one marble is selected from each box randomly what is the probability that both of them are of same colour? (10) (6) a) An intercom system master station provides music to six rooms. The probability that any room will be switched on and draw power is 0.4. When on, a room draw power of 0.5 W. Find the density and distribution function of the random variable “power delivered by the master station”. If the master station amplifier is over loaded when more than 2W is demanded then what is the probability of over load? b) Explain about conditional distribution and write down the properties of conditional distribution. (10) A Gaussian voltage random variable X has a mean value equal to zero and 2= 9. The voltage is applied to a square-law, full wave diode detector with transfer characteristic Y=5X2. Find the mean value of the output voltage b) Justify, how the density function of the sum of two statistically independent random variables is equal to the convolution of their individual density function. (10) a) State and prove Chebyshev’s inequality. b) Explain about central limit theorem. (10) (6) a) (6) (6)

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