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www.jntuworld.com Code.No: A109210401 SET-1 R09 II B.TECH – I SEM EXAMINATIONS, NOVEMBER - 2010 PROBABILITY THEORY AND STOCHASTIC PROCESS (COMMON TO ECE, ETM) Time: 3hours Max.Marks:75 Answer any FIVE questions All questions carry equal marks --1.a) b) Discuss discrete and continuous sample spaces with examples. If a box contains 75 good diodes and 25 defective diodes and 12 diodes are selected at random, find the probability that at least one diode is defective. [7+8] 2.a) b) Distinguish between discrete and continuous random variables with examples. An inspection plan calls for inspecting five chips and for either accepting each chip, rejecting each chip, or submitting it for reinspection, with probabilities of p1 = 0.70, p2 = 0.20, p3 = 0.10 respectively. What is the probability that all five chips must be reinspected? What is the probability that none of the chips must be reinspected? [6+9] 3.a) b) Calculate E[X] when X is binomially distributed with parameters n and p. The characteristic function for a Gaussian random variable X, having a mean value of 0, is Φ X (ω ) = exp( −σ X2 ω 2 / 2) [7+8] Find all the moments of X using Φ X (ω ) . 4.a) Derive the expressions for the distribution and density functions of sum of two statistically independent random variables. Find the conditional density functions for the joint distribution b) D L R O T N f X ,Y ( x, y ) = 4 xy e − ( x 5.a) b) W U 2 + y2 ) u ( x )u ( y ) J Define joint central moments for two random variables X and Y and explain the covariance of two random variables. Two random variables X and Y have means X = 1 and Y = 2, variance σ X2 = 4 and σ Y2 = 1 , and a correlation coefficient defined by V = -X + 2Y Find: i) The means, and ii) 6.a) b) [7+8] The correlation coefficients ρ XY = 0.4. New random variables W and V are W = X + 3Y ρVW of V and W. Explain the following i) Wide – sense stationary process and ii) Strict – sense stationary process. Discuss about the following ergodic process i) Mean Ergodic process. ii) Correlation ergodic process. www.jntuworld.com [6+9] [7+8]

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www.jntuworld.com Code.No: A109210401 7.a) b) SET-1 R09 Derive the relationship between the auto – correlation function and the power spectral density of a random process? Let the auto correlation function of a certain random process X(t) be given by A2 cos(ω0τ ) Rn (τ ) = 2 Obtain an expression for its power spectral density Sn( ω ). [7+8] 8.a) Explain the following i) Effective noise temperature ii) Average noise figure b) Two conductances G1 and G2 are at the same temperature 3000 K. Find the voltage power density spectrum at the terminals formed by the series combination of these conductances. [8+7] D L ******** R O W U T N J www.jntuworld.com

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www.jntuworld.com Code.No: A109210401 SET-2 R09 II B.TECH – I SEM EXAMINATIONS, NOVEMBER - 2010 PROBABILITY THEORY AND STOCHASTIC PROCESS (COMMON TO ECE, ETM) Time: 3hours Max.Marks:75 Answer any FIVE questions All questions carry equal marks --1.a) State and prove the Bayes’ theorem. b) A box with 15 transistors contains five defective ones. If a random sample of three transistors is drawn, what is the probability that all three are defective? [7+8] 2.a) b) State and prove the properties of conditional density function. The probability density function (Pdf) of the amplitude of speech waveforms is found to decay exponentially at a rate α, so the following Pdf is purposed: f X ( x) = Ce −α x ; −∞ < x < ∞ , Find the constant C. [7+8] D L 3.a) b) Calculate E[X3], if X is uniformly distributed. State and prove any four properties of characteristic function. 4.a) b) State and prove properties of the joint distribution for two random variables. Find the marginal densities of X and Y using the joint ⎡ ⎛ x ⎞⎤ FX ,Y ( x, y ) = 2u ( x)u ( y ) exp ⎢ − ⎜ 4 y + ⎟ ⎥ . 2 ⎠⎦ ⎣ ⎝ R O W U [7+8] density [7+8] 5.a) b) Explain Gaussian density function for N random variables. State and prove the properties of joint moment generating function. [7+8] 6.a) Explain the following with examples i) Discrete time stochastic process and ii) Continuous time stochastic process. Explain the first and second order stationary random processes. [8+7] b) 7.a) b) 8.a) b) T N J “The Power Spectral density of any random waveform and its autocorrelation function are related by means of Fourier transform”. Prove and illustrate the above statement. The power Spectral density of X(t) is given by ⎧1 + ω 2 for (ω ) < 1 S XX (ω ) = ⎨ otherwise ⎩ 0 Find the autocorrelation function. [8+7] Explain the following i) Available power density ii) Effective noise temperature iii) Noise figure A mixer Stage has a noise figure of 20dB and this is preceded by an amplifier that has a noise figure of 9dB and an available power gain of 15 dB. Calculate the overall noise figure referred to the input. [9+6] ******** www.jntuworld.com

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www.jntuworld.com Code.No: A109210401 SET-3 R09 II B.TECH – I SEM EXAMINATIONS, NOVEMBER - 2010 PROBABILITY THEORY AND STOCHASTIC PROCESS (COMMON TO ECE, ETM) Time: 3hours Max.Marks:75 Answer any FIVE questions All questions carry equal marks --1.a) Define the following two kinds of probability. i) Probability as a measure of frequency of occurrence ii) Probability based on an axiomatic theory. b) A coin is to be tossed until a head appears twice in a row. What is the sample space for this experiment? If the coin is fair, what is the probability that it will be tossed exactly four times? [8+7] 2.a) b) 3.a) b) State and prove any four properties of probability distribution function. Find the value of the constant k so that ⎧kx 2 (1 − x3 ), 0 < x < 1 f ( x) = ⎨ 0, otherwise ⎩ is a proper density function of a continuous random variables. D L R O [10+5] Calculate E[X] if X is a Poisson random variable with parameter λ. Show that a linear transformation of a Gaussian random variable produces another Gaussian random variable. [7+8] W U 1 − ( x /6+ y /3) e for x ≥ 0, y ≥ 0 18 Show that X and Y are statistically independent random variables. Discuss the Central limit theorem. [8+7] 5.a) b) State and prove the properties of covariance function. State and prove the properties of joint moment generating function. [7+8] 6.a) b) What is the importance of covariance function? State and prove the properties of auto correlation function. 7.a) Derive the relationship between power spectral density of input and output random process of an LTI system. A random process X(t) whose mean value is 2 and auto correlation function e−2 τ 1 RXX (τ ) = 4 is applied to a system whose transfer function is . Find 2 + jw the mean value, auto correlation, power density spectrum and average power of output signal Y(t). [6+9] 4.a) b) b) The joint Pdf is f X ,Y ( x, y ) = T N J www.jntuworld.com [5+10]