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- Probability and Random Processes - PRP
- 2017
- PYQ
**Biju Patnaik University of Technology Rourkela Odisha - BPUT**- Electronics and Communication Engineering
- B.Tech
**2253 Views**- 30 Offline Downloads
- Uploaded 1 year ago

Registration No: Total Number of Pages: 03 B.Tech PET3D001 3rd Semester Regular/Back Examination 2017-18 Probability & Random Process BRANCH: Time: 3 Hours Max Marks: 100 Q.CODE: B1236 Answer Part-A which is compulsory and any four from Part-B. 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) j) Part – A (Answer all the questions) Answer the following questions: multiple type or dash fill up type Define joint probability. If A1 and A2 are equally likely, mutually exclusive and exhaustive and PB / A1 0.2, PB / A2 0.3 .find P A1 / B . If P( B) 1 , then P( A / B) .................... . If S1 corresponding to flipping a coin, then S 1 H , T , where H is the element “heads” and T represent the “tails”. Let S 2 1,2,3,4,5,6 corresponding to rolling a single die. The combined sample space S S 1 S 2 ................ Define Gaussian random variable. Find the mean and variance of the random variable X, where f (x) is the uniform distribution on 0,10 . The conditional distribution function is _______than the ordinary distribution function. Define equal and unequal distribution. Define jointly Gaussian random variables. If we draw a card from well-shuffled pack of 52 cards,what is the probability that the card is either an ace or a king? Answer the following questions: Short answer type Define conditional probability. If A and B are events such that A B , show that P A PB . Write the properties of distribution function. Define probability density function. Define marginal distribution . State the Rayleigh density and distribution function. Define exponential distribution. State the moment generating function of a random variable X about the origin. Find the value of k, f ( x) K ( x x 2 ) for 0 x 1 , f ( x) 0 otherwise. State central limit theorem. (2 x 10) (2 x 10)

Part – B (Answer any four questions) Q3 a) State and prove Baye’s Rule b) Two persons are competing for the post of the principal of a college. The probabilities that the first and second persons 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? (10) (5) Q4 a) Use De morgan’s laws show that __________________ ___ ___ ___ ___ (i) A ( B C ) ( A B ) A C (10) __________________ ___ ___ __ (ii) A ( B C ) ( A B C ) . b) The number of telephone calls received in an office during lunch hour has probability function given below. No.of calls:x 0 1 2 3 4 5 6 Probability:P(x) 0.05 0.20 0.25 0.20 0.15 0.10 0.05 (i)Find the probability that there will be 3 or more calls. (ii)Find the probability that there will be an odd number of calls. Q5 a) A candy company distributes boxes of chocolates with a mixture of creams, toffees and nuts coated in both light and dark chocolate. For a randomly selected box, let X and Y, respectively, be the proportions of the light and dark chocolates that are creams and suppose that the joint 2 density function is f ( x, y ) 2 x 3 y ,0 x 1,0 y 1 f ( x, y) 0 5 elsewhere 1 1 1 (i)Find P X , Y A, where A is the region ( x, y) | 0 x , y . 2 4 2 (ii)Verify random (10) f ( x, y ) dxdy 1 . b) Assume that the height of clouds above the ground at some location is a Gaussian random variable X with a x 1830m and x 460m .Find the probability that clouds will be higher than 2750 m . Q6 a) Two (5) variables X &Y have joint density function xy ,0 x 1,0 y 2 f XY x, y 0 otherwise. 3 Show that X & Y are not independent. Find the conditional density functions. Check whether the conditional density function are valid. b) If the probability of hitting a target is 25% and 4 shots are fired independently, what is the probability that the target will be hit at least once? (5) (10) f XY x, y x 2 (5)

Q7 a) Show that the distribution for which the characteristic function is e 1 1 has the density function f ( x) , x . 1 x 2 b) Find the moment generating function of the random variable X having 1 f ( x) 0 the probability density function f ( x ) , 1 x 2 K otherwise. (10) Q8 a) State and prove Chebyshev’s inequality b) Find the density W X Y where the densities X & Y are assumed to 1 1 be f X ( x) u ( x) u ( x a) , f Y ( y ) u ( y ) u ( y b) with 0 a b . a b (10) (5) Q9 a) Define Poisson distribution, in a certain industrial facility accident occurs infrequently. It is known that the probability of an accident on any given day is 0,005 and accidents are independent of each other. (i)What is the probability that in any given period of 400 days there will be an accident on one day? (ii)What is the probability that there are at most 3 days with an accident? b) Let X be normal with mean 10 and variance 4. Find P( X 12), P(9 X 13), P( X 11) . (10) (5) (5)

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