Rr221101-probability-theory--and--stochastic-process

Set No. 1

Code No: RR221101

II B.Tech II Semester Supplimentary Examinations, Aug/Sep 2008 PROBABILITY THEORY AND STOCHASTIC PROCESS (Bio-Medical Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) Define Probability density function and obtain the relationship between probability and probability density. (b) Consider the probability density f (x) = ae−b|x| where x is a random variable Whose allowable values range from x = −∞to∞. Find i. the CDF F (x) ii. the relationship between a and b. and iii. the probability that the out come x lies between 1 and 2.

[7+9]

2. (a) Derive an expression for, the error function of the standard normal Random variable (b) Lifetime of IC chips manufactured by a semiconductor manufacturer is approximately normally distributed with mean = 5x 106 hours and standard deviation of 5x 105 hours. A mainframe manufacturer requires that at least 95% of a batch should have a lifetime greater than 4x106 hours. Will the deal be made? [8+8] 3. (a) Find the moment generating function of the random variable having probability density function fX (x) = x, 0≤x≤1 = -2-x, 1≤x≤2 = 0, else where (b) Find the moment generating function of the random variable whose moments are mr = (r + 1)!2r . [8+8] 4. A class of modulation signal is modulated by Xc (t) = AX(t)Cos(̟c t+θ) Where x(t) is the message signal and A Cos (̟c t + θ) is the carrier. The message Signal x(t) is modeled as a zero mean stationary random process with the autocorrelation function Rxx (τ ) and the PSD Gx (f ). The carrier amplitude A and frequency ̟c are assumed to be constants and the initial carrier phase θ is assumed to be a random variable uniformly distributed in the interval (−Π, Π). Further more x(t) and θ are assumed to be independent. (a) Show that Xc (t) is stationary (b) Find the PSD of Xc (t).

[8+8] 1 of 2

Set No. 1

Code No: RR221101

5. (a) Derive the relation between PSDs of input and output random process of an LTI system. (b) X(t) is a stationary random process with zero mean and auto correlation 1 RXX (τ ) e−2|τ | is applied to a system of function H (w) = jw+2 Find mean and PSD of its output. [8+8] 6. (a) What are the causes of thermal noise? (b) What are the causes of shot noise?

[8+8]

7. In TV receivers, the antenna is often mounted on a tall mask and a long lossy cable is used to connect the antenna and receiver. To overcome the effect of noisy cable, a preamplifier is mounted on the antenna. The parameters of the different stages are Preamplifier gain = 20 dB Preamplifier Noise figure = 6 dB Lossy cable noisy figure = 3 dB Cable Loss = -20 dB Receiver front end gain = 60 dB Receiver Noise figure = 16 dB Determine the overall noise figure of the system. [16] 8. (a) A code is composed of dots and dashes. Assume that a dash is three times as long as the dot and has one-third the probability of occurrence. Find, i. The information in a dot and that in a dash, and ii. The entropy in the dot - dash code. (b) Suppose 100 voltage levels are employed to transmit 100 equally likely messages. Assume the system to be a Gaussian channel with λ = 3.5 and bandwidth B = 104 Hz. Find S/N. [8+8] ⋆⋆⋆⋆⋆

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Set No. 2

Code No: RR221101

II B.Tech II Semester Supplimentary Examinations, Aug/Sep 2008 PROBABILITY THEORY AND STOCHASTIC PROCESS (Bio-Medical Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. A binary communication channel carries data as one of the two types of signals denoted by 0 and 1. Owing to noise a transmitted 0 is sometimes received as 1 and a transmitted 1 is sometimes received as a 0. For a given channel, assume a probability of 0.94 that a transmitted 0 is correctly received as a 0 and a probability of 0.91 that a transmitted 1 is received as a 1.Further assume a probability of 0.45 of transmitting a 0. If a signal is sent, Determine (a) probability that a 1 is received. (b) Probability that a 0 was received (c) Probability that a 1 was transmitted, given that a 1 was received (d) Probability that a 0 was transmitted, given that a 0 was received (e) Probability of as error

[3+3+4+4+2] −

−

2. (a) If A and B are independent events, prove that the events A and B, A and B; and A and B are also independent. [6] (b) A1, A2 and A3 are three mutually exclusive and exhaustive sets of events associated with a random experiment E1. Events B1,B2 and B3 are mutually exclusive and exhaustive sets of events associated with a random experiment E2. The joint Probabilities of occurrence of these events and some marginal probabilities are listed in the table given below: B1 B2 B3 A1 3/36 * 5/36 A2 5/36 4/36 5/36 A3 * 6/36 * P(Bj) 12/36 14/36 * i. Find the missing probabilities (*) in the table. ii. Find P(B3|A1) and P(A1|B3) iii. Are events A1 and B1 statistically independent?

[4+4+2]

3. (a) If X and Y are two random variables which are Gaussian. If a random variable Z is defined as Z=X+Y, Find fZ (Z). (b) Prove that the characteristic function and probability density function form a fourier transform pair. [8+8]

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Set No. 2

Code No: RR221101

4. (a) Prove that PSD and Auto correlation function of Random process form a fourier transform pair. (b) A random process has the power density spectrum Sxx (̟) = Find the average power in the process.

6ω 2 1+ω 4

[8+8]

5. (a) Find the PSD of a random process z(t) = X(t) + y(t) where x(t) and y(t) are zero mean, individual random process. (b) A wss random process x(t) is applied to the input of an LTI system whose impulse response is 5t.e−2t The mean of x(t) is 3. Find the output of the system. [8+8] 6. (a) What are the characteristics of White noise? (b) Discuss the spectral distribution of thermal noise.

[8+8]

7. (a) What are the precautions to be taken in cascading stages of a network in the point of view of noise reduction? (b) What is the need for band limiting the signal towards the direction increasing SNR. [8+8] 8. (a) State the axioms of an entropy function. Show that only function which satisfies the above axiom is, n P H (P1 , P2 , .........Pn ) = λ pi log2 pi i=1

(b) A man is informed that when a pair of dice was rolled, the result was a seven. How much information is there in this message? [8+8] ⋆⋆⋆⋆⋆

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Set No. 3

Code No: RR221101

II B.Tech II Semester Supplimentary Examinations, Aug/Sep 2008 PROBABILITY THEORY AND STOCHASTIC PROCESS (Bio-Medical Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) The number of times that an electric switch operate before having to be discarded is found to be a random variable with probability mass function (pmf): p(x) = A(1/3)x for x = 0, 1, 2,...... =0 otherwise. i. Find the value of A that makes p(.) a p.m.f. ii. Sketch the p.m.f. iii. What is the probability that the number of times the switch will operate before having to be discarded is greater than 5, an even number (regard 0 as even.), and an odd number. (b) A continuous random variable has the p.d.f. given by fx (x) = 5 − Cx; 0 ≤ x ≤ 5. If Y = ax2 +b, find the p.d.f.of Y.

[8 + 8]

2. (a) The number of newspapers that a certain delivery boy is able to sell in a day is found to be a numerical valued random phenomenon, with a probability function specified by the p.m.f.p(.) given by: p(x) = Ax

x = 1, 2, 3, ..................., 50

= A(100 − x)

x = 51, 52, ..............100

= 0 otherwise i. Find the value of A that makes p (.) a p.m.f. and sketch its graph. ii. What is the Probability that the number of newspapers sold tomorrow is A. more than 50 B. less than 50 C. equal to 50 D. between 25 and 75 exclusive E. an odd number. iii. Let the events indicated in (ii) be denoted respectively by A, B, C, D and E. Find P(A|B), P(A|C), P(A|D)andP(C|D). Are A and B independent? Are A and D independent? Are C and D independent events? [4+5+7] 3. (a) The joint probability density function of random variables X and Y is f (x, y) = 1 4

′

′

′

′

exp (|x − y|) − α < x < α , and − α < y < α If another random variable ‘Z’ is defined such that, Z = X + Y Find fz (Z). 1 of 2

Set No. 3

Code No: RR221101

(b) Two random variables x and y have the following joint probability density function f(x,y) = 2-x-y, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 = 0, other wise Find i. Marginal probability density functions of x and y ii. Var (x) and Var (y)

[8+8]

4. (a) Which are the following are suitable auto correlation functions? i. Acos̟0 τ ii. AΠ(τ /τ0 ) where Π(x) is a unit area rectangular function (b) Suppose we are given a cross power spectrum defined by Sxx (̟) = a + ((jb̟/w)); −W < ̟ < W = 0 : Elsewhere Where W > 0, a and b are real constants. Find the cross correlation function. [8+8] 5. (a) Find the PSD of a random process z(t) = X(t) + y(t) where x(t) and y(t) are zero mean, individual random process. (b) A wss random process x(t) is applied to the input of an LTI system whose impulse response is 5t.e−2t The mean of x(t) is 3. Find the output of the system. [8+8] 6. (a) Write short notes on i. Noise power spectral density. ii. Noise suppression in semiconductor devices. (b) Discuss the role of temperature variation in the operation of electronic devices in the point of view of noise generation. [5+5+6] 7. (a) Discuss the significance of noise equivalent temperature of an electronic system. (b) Evaluate the equivalent noise temperature of a two port device with a matched source and a matched load. [8+8] 8. (a) Messages Q1 , Q2 , Q3 have probabilities P1 , P2 , P3 such that P1 + P2 + P3 = 1. Find Hmax by first eliminating P3 . (b) Consider five messages have probabilities 1/2, 1/4, 1/8, 1/16, 1/16. Using Shannon - Fano algorithm, develop efficient code. For that code find the average number of bits message and compare with H. [8+8] ⋆⋆⋆⋆⋆

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Set No. 4

Code No: RR221101

II B.Tech II Semester Supplimentary Examinations, Aug/Sep 2008 PROBABILITY THEORY AND STOCHASTIC PROCESS (Bio-Medical Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) State and prove Bayes theorem of probability. (b) In a single throw of two dice, what is the probability of obtaining a sum of at least 10? [8+8] 2. (a) Explain the Rayleigh probability density function. (b) Find the mean value, the mean squared value and the cumulative distribution function for the Rayleigh distribution with parameter α> 0, specified by the pdf [8+8] x f (x) = 2 exp α

  1 x − 2 α

3. (a) Find the rth moment of the random variable X in terms of its characteristic function φx (̟). 2 (b) Prove that σx+y = σx2 + σy2 + 2[E[XY]]-E[X]E[Y]] where X and Y are two random variables. [8+8]

4. A class of modulation signal is modulated by Xc (t) = AX(t)Cos(̟c t+θ) Where x(t) is the message signal and A Cos (̟c t + θ) is the carrier. The message Signal x(t) is modeled as a zero mean stationary random process with the autocorrelation function Rxx (τ ) and the PSD Gx (f ). The carrier amplitude A and frequency ̟c are assumed to be constants and the initial carrier phase θ is assumed to be a random variable uniformly distributed in the interval (−Π, Π). Further more x(t) and θ are assumed to be independent. (a) Show that Xc (t) is stationary (b) Find the PSD of Xc (t).

[8+8]

5. (a) Derive the relation between PSDs of input and output random process of an LTI system. (b) X(t) is a stationary random process with zero mean and auto correlation 1 RXX (τ ) e−2|τ | is applied to a system of function H (w) = jw+2 Find mean and PSD of its output. [8+8] 6. (a) Explain available power of a noise source. 1 of 2

Set No. 4

Code No: RR221101

(b) Explain the components of noise power spectral density.

[8+8]

7. (a) Bring out the difference between narrowband and broadband noises (b) Describe the quadrature representation of narrowband noise.

[8+8]

8. (a) A code is composed of dots and dashes. Assume that a dash is three times as long as the dot and has one-third the probability of occurrence. Find, i. The information in a dot and that in a dash, and ii. The entropy in the dot - dash code. (b) Suppose 100 voltage levels are employed to transmit 100 equally likely messages. Assume the system to be a Gaussian channel with λ = 3.5 and bandwidth B = 104 Hz. Find S/N. [8+8] ⋆⋆⋆⋆⋆

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