Rr210403 Probability Theory Stochastic Process Supply 2006

Set No. 1

Code No: RR210403

II B.Tech I Semester Supplementary Examinations, March 2006 PROBABILITY THEORY & STOCHASTIC PROCESS ( Common to Electronics & Communication Engineering and Electronics & Telematics) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) Define the following and give one example for each. i. ii. iii. iv.

Sample space Event Mutually exclusive events. Collectively exhaustive events. [4x2]

(b) In three boxes, there are Capacitors as shown in the following table: Value in µf

Number in box

↓

→ 1

2

3

1.0

70

80

145

0.1

55

35

75

0.01

20

95

25

An experiment consists of first randomly selecting a box (assume that each box has the same probability of selection) and then randomly selecting a capacitor from the chosen box. i. What is the probability of selecting 0.01uf capacitor, given that the box2 is chosen? ii. If a 0.01 µf capacitor is chosen, what is the probability that it came from the second box [4+4] 2. Two discrete random variables X and Y have joint p.m.f. given by the following table X ↓ 1 2 3 Y ← 1 1/12 1/6 1/12 2 1/6 1/4 1/12 3 1/12 1/12 0 1 of 3

Set No. 1

Code No: RR210403 Compute the probability of each of the following events (a) X ≤ 11/2 (b) XY is even (c) Y is even given that X is even.

[5+5+6] 3. (a) For a function Y=(X − mx )/σx , prove that mean is zero & variance is 1 (b) For the joint distribution of (X,Y) given by fxy (x, y) = 4a12 [(1 + xy) (x2 − y 2] , |x| <= a, |y| <= a, a > 0 = 0, otherwise Show that the Characteristic function of X+Y is equal to the product of the characteristic function of X & Y. [8+8] 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. Find the input auto correlation function, output autocorrelation and o/p spectral density of RC low pass filter, where the filter is subjected to a white noise of spectral density N0 /2. [16] 6. Give reasons for the following: (a) In any communication system the first stage must have low noise operation. (b) Describe how FET gives low noise performance compared to BJT. [8+8] 7. (a) Show that the effective noise temperature of n networks in cascade is given by, Te = Te1 + Te2 /g1 + Te3 /g1 g2 + ................... + Ten /g1 g2 gn−1 (b) A low noise receiver for satellite ground station consists of the following stages Antenna with Ti = 125K Waveguide with a loss of 0.5dB Power amplifier with ga = 30dB, Te = 6K, BN = 20 MHz TWT amplifier with ga = 16dB, F = 6dB, BN = 20 MHz Calculate the effective noise temperature of the system. [8+8] 8. (a) Compare discrete and continuous channel with respect to information transmission.

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Set No. 1

Code No: RR210403

(b) Derive an expression for channel capacity of a continuous channel in the presence of White Gaussian noise. [8+8] ⋆⋆⋆⋆⋆

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Set No. 2

Code No: RR210403

II B.Tech I Semester Supplementary Examinations, March 2006 PROBABILITY THEORY & STOCHASTIC PROCESS ( Common to Electronics & Communication Engineering and Electronics & Telematics) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) If A and B are any events, not necessarily mutually exclusive events, derive an expression for probability of A Union B. When A and B are mutually exclusive, what happens to the above expression derived? (b) Define the term Independent events. State the conditions for independence of i. any two events A and B. ii. any three events A, B and C. (c) A coin is tossed. If it turns up heads, two balls will be drawn from box A, otherwise, two balls will be drawn from box B. Box A contains three black and five white balls. Box B contains seven black and one white balls. In both cases, selections are to be made with replacement. What is the probability that Box A is used, given that both balls drawn are black? [5+6+5] 2. (a) Explain the Gaussian distribution with a neat sketches of pdF and cdF. (b) An analog signal received at the detector (measured in microvolts) may be modeled as a Gaussian random variable N (200,256) at a fixed point in time. What is the probability that the signal will exceed 240 µ vs. what is the probability that the signal is larger than 240 µ V, given that it is larger than 210 µ Vs? [8+8] 3. (a) Prove that mean is ‘m’ and variance is σ 2 for Gaussian distribution function. (b) Find the moment generating and Characteristic function of the random variable X which has uniform distribution. [8+8] 4. Find the Auto correlation function and power spectral density of the Random process. x(t) = K Cos (̟o t + θ) Where θ is a Random variable over the ensemble and is uniformly distributed over the Range ( 0, 2π) [16] 5. Find the input auto correlation function, output autocorrelation and o/p spectral density of RC low pass filter, where the filter is subjected to a white noise of spectral density N0 /2. [16] 1 of 2

Set No. 2

Code No: RR210403

6. (a) Explain how partition noise is present in electron devices? (b) Explain the usefulness of knowing the noise power spectral density of a network. [8+8] 7. (a) The noise figure of an amplifier at room temperature (T=29.K) 0.2db. Find the equivalent temperature. (b) Explain the concept of effective input noise temperature [8+8] 8. (a) For a system the bandwidth is 4 KHz and S/N ratio is 14. If the bandwidth is increased to 5 KHz, find the required S/N ratio to have the same channel capacity and find the percentage change in signal power. (b) Using the definition of H (x) and H (X/y), Show that, I (x, y) =

P x,y

P (x,y) P (x, y) log P (x),P (y)

[6+10] ⋆⋆⋆⋆⋆

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Set No. 3

Code No: RR210403

II B.Tech I Semester Supplementary Examinations, March 2006 PROBABILITY THEORY & STOCHASTIC PROCESS ( Common to Electronics & Communication Engineering and Electronics & Telematics) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) State and derive the theorem on Total probability. (b) A shipment of components consists of three identical boxes. One box contains 2000 components of which 25% are defective, the second box has 5000 components of which 20% are defective and the third box contains 2000 components of which 600 are defective. A box is selected at random and a component is removed at random from the box. What is the probability that this component is defective? What is the probability that it came from the second box? [8+8] 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) For the random variable X whose density function is f (x) =

1 , a≤x≤b b−a 0, otherwise

Determine i. Moment generating function ii. Mean and Variance (b) Prove that E(X) = E(X/Y), where X and Y are two random variables [8+8] 4. (a) Explain Ergodic random process (b) State and prove properties of Auto correlation function [8+8]

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Set No. 3

Code No: RR210403

5. White noise n(t) with G(f) =η/2 is passed through a low pass RC network with a 3dB frequency fc . (a) Find the autocorrelation R(τ ) of the out put noise of the network. (b) Sketch P(τ ) =R(τ )/R(0) (c) Find ̟c (τ )such that P (τ ) ≤0.1. [8+4+4] 6. (a) Explain how the available noise power in an electronic circuit can be estimated. (b) What are the different noise sources that may be present in an electron devices? [8+8] 7. (a) Define noise temperature and noise figure. (b) Show that the noise figure of a well designed receiver is usually near about the noise figure of the low noise RF stage amplifier at the antenna input. [8+8] 8. (a) Describe the channel capacity of a discrete channel. (b) Explain Shannon - Fano algorithm to develop a code to increase average information per bit. [8+8] ⋆⋆⋆⋆⋆

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Set No. 4

Code No: RR210403

II B.Tech I Semester Supplementary Examinations, March 2006 PROBABILITY THEORY & STOCHASTIC PROCESS ( Common to Electronics & Communication Engineering and Electronics & Telematics) 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) Define Conditional Probability mass function. (b) If two random variables have the joint probability density 2 f(x1 , x2 ) = (x1 + 2x2 ) 3 = 0 else where

for 0 < x1 < 1, 0 < x2 < 1

Find i. the marginal density of x2 . ii. Conditional density of the first given that the second takes on the value x2 . (c) A pair of dice is tossed. Define a random variable X to be the difference of the face values turned up. Determine the probability mass function of X [4+6+6] 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). (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 1 of 2

Set No. 4

Code No: RR210403

i. Marginal probability density functions of x and y ii. Var (x) and Var (y) [8+8] 4. (a) Explain Ergodic random process (b) State and prove properties of Auto correlation function [8+8] 5. Find the input auto correlation function, output autocorrelation and o/p spectral density of RC low pass filter, where the filter is subjected to a white noise of spectral density N0 /2. [16] 6. (a) Explain how partition noise is present in electron devices? (b) Explain the usefulness of knowing the noise power spectral density of a network. [8+8] 7. (a) Show that the effective noise temperature of n networks in cascade is given by, Te = Te1 + Te2 /g1 + Te3 /g1 g2 + ................... + Ten /g1 g2 gn−1 (b) A low noise receiver for satellite ground station consists of the following stages Antenna with Ti = 125K Waveguide with a loss of 0.5dB Power amplifier with ga = 30dB, Te = 6K, BN = 20 MHz TWT amplifier with ga = 16dB, F = 6dB, BN = 20 MHz Calculate the effective noise temperature of the system. [8+8] 8. (a) A source emits six messages with probabilities 1/2, 1/4, 1/8, 1/16, 1/32 and 1/32. Determine average information per message. What would be the information if the above messages were equally probable? (b) Distinguish between the terms “Self Information and Mutual Information”. [8+8] ⋆⋆⋆⋆⋆

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