Code: 15A04304

B.Tech II Year I Semester (R15) Regular Examinations November/December 2016

PROBABILITY THEORY & STOCHASTIC PROCESSES (Electronics and Communication Engineering)

Time: 3 hours Max. Marks: 70 PART – A

(Compulsory Question)

***** 1 Answer the following: (10 X 02 = 20 Marks) (a) What is the condition for a function to be a random variable? (b) Define Gaussian random variable. (c) How interval conditioning is different from point conditioning? (d) When N random variables are said to be jointly Gaussian? (e) Explain about strict-sense stationery processes. (f) Where the Poisson random processes is used? Explain.

(g) Examine the function 33 26

2

++ ωωω

for valid PSD.

(h) Correlate CPSD and CCF. (i) Analyze the power density spectrum of response. (j) List the properties of band limited processes.

PART – B

(Answer all five units, 5 X 10 = 50 Marks)

UNIT – I

2 (a) Give Classical and Axiomatic definitions of Probability. (b) In a single through of two dice, what is the probability of obtaining a sum of at least 10? OR

3 (a) What is the concept of Random Variable? Explain with a suitable example. (b) A random variable X has the distribution function:

∑=

−=12

1

2

)(650

)(n

X nxunxF

Find the probabilities (i) P-∞< X ≤ 6.5. (ii) PX > 4 (iii) P6 < X ≤ 9.

UNIT – II

4 (a) State and explain the central limit theorem. (b) Given the function:

.elsewhere 0,

3y3- ,2x2- ,y)b(xy)(x,f2

XY

<<<<+=

(i) Find a constant ‘b’ such that this is a valid density function. (ii) Determine the marginal density functions fx(x) and fy(y).

OR 5 (a) What are the properties of Jointly Gaussian Random variables? (b) A random variable X has .2,11,3 22 ==−= xandXX σ For a new random variable Y= 2X-3, find:

(i) (ii) (iii)

Contd. in page 2

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R15

Code: 15A04304

UNIT – III

6 (a) List and explain various properties of Autocorrelation function. (b) Given the Autocorrelation function of the processes:

2XX

61

425)(Rτ+

+=τ

Find the mean and variance of the process X(t). OR

7 (a) Compare the Cross Correlation Function with Autocorrelation function. (b) Assume that an Ergodic random process X(t) has an autocorrelation function:

)]12cos(41[6

218)(R2XX τ+τ+

+=τ

(i) Find X . (ii) Does this process have periodic component? (iii)What is the average power in X(t)?

UNIT – IV

8 (a) State and explain the Wiener-Khintchine relation. (b) Obtain the auto correlation function corresponding to the power density spectrum:

2

)9(

8) (ωS2

XXω+

=

OR 9 (a) Define Power Spectral Density? List out its properties.

(b) Compute the average power of the process having power spectral density 4

2

16ωω+

.

UNIT – V

10 (a) What is LTI system? How the response can be obtained from LTI system. (b) Find the system response, when a signal x(t)= u(t) e-2t is applied to a network having an impulse

response h(t) = 3u(t) e-3t. OR

11 (a) Explain about mean and mean square value of system response? (b) A random process X(t) is applied to a network with impulse response: h(t) = u(t) t e-3t. The cross

correlation of X(t) with the output Y(t) is known to have the same form RXX(τ) = u(τ) τ e-3τ. (i) Find the autocorrelation of Y(t). (ii)What is the average power in Y(t)

*****

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R15

Code: 13A04304

B.Tech II Year I Semester (R13) Regular & Supplementary Examinations December 2015

PROBABILITY THEORY & STOCHASTIC PROCESSES (Electronics and Communication Engineering)

Time: 3 hours Max. Marks: 70

PART – A (Compulsory Question)

***** 1 Answer the following: (10 X 02 = 20 Marks) (a) What are the conditions to be satisfied for the statistical independence of three events A, B and C? (b) Show that . (c) Two random variables X and Y have the following values:

Find the correlation coefficient.

(d) Define the joint moments about the origin. (e) Define WSS random process. (f) Determine the mean-square value of a random process with autocorrelation function: (g) A random process has the power density spectrum Find the average power in the

process. (h) Define rms bandwidth of the power spectrum. (i) Impulse response of a linear system is The input to this system is a

sample function from a random process having an autocorrelation function of Find the autocorrelation of the output.

(j) A stationary random process with a mean of 2 is passed through an LTI system with Determine the mean of the output process.

PART – B

(Answer all five units, 5 X 10 = 50 Marks)

UNIT – I

2 (a) Define conditional distribution and density functions and list their properties.

(b) A continuous random variable X has a PDF Find ‘a’ and ‘b’ such that:

(i) . (ii) . OR

3 (a) Define random variable and give the concept of random variable with an example. (b) The probability density function of a random variable has the form , where is

the unit step function. Find the probability that

UNIT – II

4 (a) Define marginal density and distribution functions. (b) Let X and Y be jointly continuous random variables with joint probability density function:

Find: (i) (ii) . (iii) Are X and Y independent?

OR 5 (a) State central limit theorem for the following two cases:

(i) Equal distributions. (ii) Unequal distributions. (b) Let and zero elsewhere. Find .

Contd. in page 2

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R13

Code: 13A04304

UNIT – III

6 (a) A random process has sample functions of the form:

Where A is a random variable uniformly distributed from 0 to 10. Find the autocorrelation function of this process.

(b) Show that OR

7 (a) Show that the autocorrelation function of a stationary random process is an even function of (b) Give the classification of random processes.

UNIT – IV

8 (a) A stationary random process has a two-sided spectral density given by:

Find the mean-square value of the process if (b) List the properties of power spectral density function. OR 9 (a) For two jointly stationary random processes, the cross-correlation function is . Find

the two cross-spectral density function. (b) List the properties of cross power spectral density function.

UNIT – V

10 Show that OR

11 Write short notes on the following: (a) Bandpass random process. (b) Band-limited random process.

*****

Page 2 of 2

R13

Code: 9A04303 B.Tech II Year I Semester (R09) Regular & Supplementary Examinations December/January 2013/14

PROBABILITY THEORY AND STOCHASTIC PROCESSES (Common to EIE, E.Con.E & ECE)

Time: 3 hours Max Marks: 70 Answer any FIVE questions

All questions carry equal marks *****

*****

1 (a) Define the following with example: (i) Sample space. (ii) Event. (iii) Mutually exclusive event. (iv) Independent event. (v) Exhaustive event.

(b) When a die is tossed find the probabilities of the event A = odd number shown up; B = Number larger than 3 shown up then find out A B and A B.

2 (a) Define probability density function and explain with an example and write its properties. (b) The random variable has following density function:

(i) Find value of k. (ii) P ( . 1 2 3 4 5 6 7 k 2k 2k 3k

3 (a)

The density function of a random variable X is:

(i) E [x], (ii) E [(x-1)2], (iii) E [3x -1]. (b) For Poisson distributions find out moment generating function and characteristic function. 4 (a) Distinguish between joint distribution and marginal distribution. (b)

Joint probability density function of two random variables X and Y.

Find: (i) Value of ‘a’; (ii) . 5 (a) Explain relation between marginal and joint characteristic function. (b) In a control system a random voltage is known to have mean value and a second

moment If the voltage x is amplified by an amplifier that gives an output 2, 2 and

6 (a) State the conditions for wide sense stationary random process. (b) Explain the classifications of random process. 7 (a) What is an ergodic random process, present the necessary expression to support the argument? (b) Consider a random process where is a random variable

uniformly distributed over where is any real number find 8 (a) Explain the relation between power spectrum and auto correlation function of random process. (b) Write any two properties of cross power density spectrum.

1

Code: 9A04303 B.Tech II Year I Semester (R09) Regular & Supplementary Examinations December/January 2013/14

PROBABILITY THEORY AND STOCHASTIC PROCESSES (Common to EIE, E.Con.E & ECE)

Time: 3 hours Max Marks: 70 Answer any FIVE questions

All questions carry equal marks *****

1 (a) State and prove Baye’s theorem of probability. (b) Two similar boxes ‘A’ and ‘B’ contain 2 white and 3 red balls 4 white and 5 red balls respectively. If

a ball is selected at random from one of the boxes, then find the probability that the box is ‘B’ when the ball is red.

2 (a) Write the method for defining conditional event. (b)

Check whether following is a probability density function or not

3 (a) Explain variance and skew. (b) The mean and variance of binomial distribution are 4 and 4/3 respectively find (c) Find the expected value of the number on a die when thrown.

4 (a) Define and explain the conditional properties. (b)

The joint probability density function of two random variables x and y given by

(i) Find the value of c. (ii) Marginal distribution function X and Y.

5 (a) Write the expression for expected value of a function of random variables and prove that the mean

value of a weighted sum of random variables equals the weighted sum of mean values. (b) Two Gaussian random variables X1 and X2 defined by the mean and co-variance matrices

two new random variable s Y1 and Y2 are formed using the transformation

. Find the matrices , and also find the torrelation co-efficient of and i.e., .

6 (a) Explain the types of statistical averages. (b) Define a random process where ‘A’ is Gaussian random variable with zero mean

and variance 2 is X(t) is stationary in any sense. 7 (a) Define cross correlation and write its properties. (b) A random process is defined as where is a uniform random variable over

(0, 2 Verify the process is ergodic in the mean sense and auto correlation sense. 8 (a) Write the relation between cross power spectrum and cross correlation function. (b) If power spectral density of a random process is given by

Find the auto correlation function.

*****

2

Code: 9A04303 B.Tech II Year I Semester (R09) Regular & Supplementary Examinations December/January 2013/14

PROBABILITY THEORY AND STOCHASTIC PROCESSES (Common to EIE, E.Con.E & ECE)

Time: 3 hours Max Marks: 70 Answer any FIVE questions

All questions carry equal marks *****

Contd. in Page 2

Page 1 of 2

1 (a) What is sample space? Explain the discrete and continuous sample space. (b) In a box there are 100 resistors having resistances and tolerances as shown in the table.

Resistance Tolerance

Total 5% 10%

22 10 14 24 47 28 16 44 100 24 8 32 Total 62 38 100

Let a resistor be selected from the box and assume each resistor are equally likely occur. Define three events ‘A’ as draw a 47 Ω resistor, ‘B’ as draw a resistor with 5% tolerance and ‘C’ as draw a 100 Ω resistor. (i) The probability of drawing a 47 Ω resistor given that the resistor drawn is 5%. (ii) The probability of drawing a 47 Ω resistor given that the resistor drawn is 100 Ω. (iii) The probability of drawing a resistor of 5% tolerance given resistor is 100 Ω. (iv) Find remaining conditional probabilities.

2 (a) Sketch probability density function and probability distribution function of: (i) Exponential distribution. (ii) Rayleigh distribution. (iii) Uniform distribution.

(b) Define conditional distribution function and write their properties. 3 (a) State and prove any three properties of variance of a random variable. (b) For binomial density prove:

(i) . (ii) . (iii) 2 = npq. 4 (a) Define and explain joint distribution function and joint density function of two random variables X

and Y. (b) The joint space for two random variable X and Y and its corresponding probabilities are shown in

table 1, 1 2, 2 3, 3 4, 4

0.2 0.3 0.35 0.15 Find and plot (a). Marginal distribution of X and Y.

(c) (d) .

3

Code: 9A04303

*****

Page 2 of 2

5 (a) Show that the variance of a weighted sum of uncorrelated random variables equals the weighted sum of the variance of the random variables.

(b)

Two random variables X and Y have the density function:

(i) Find all the order moment. (ii) Find covariance. (iii) X and Y and uncorrelated.

6 (a) Explain the concept of random process. (b) Distinguish between:

(i) Deterministic random process and non-deterministic random process. (ii) Stationary and non- stationary random process.

7 (a) S.T Where A is normal random variable with zero mean and unity

variance and uniformly distributed is Assume A and are independent random variable.

(b) Discuss Gaussian random process and state its properties. 8 (a) State and explain four properties of power density spectrum of a random process. (b) If find the spectral density function, where a and b are constant.

3

Code: 9A04303 B.Tech II Year I Semester (R09) Regular & Supplementary Examinations December/January 2013/14

PROBABILITY THEORY AND STOCHASTIC PROCESSES (Common to EIE, E.Con.E & ECE)

Time: 3 hours Max Marks: 70 Answer any FIVE questions

All questions carry equal marks

1 (a) Give the classical and axiomatic definitions of probability. (b) A card is drawn from a well shuffled pack of playing cards. What is the probability that in either a spade or

ace? (c) What is the probability: (i) Leap year selected at random will contain 53 Sundays?

(ii) Non leap year selected at random will contain 53 Sundays? 2 (a) Define random variable and explain types of random variable. (b) In an experiment of rolling a die and flipping a coin the random variable ( is chosen such that.

(i) A coin tail (H) outcome corresponds to positive value of that are equal in magnitude to twice the number that shown on die. Map the elements of random variable into points on the real line and explain.

3 (a) Write about transformation. (b) Let x be a random variable defined by density function:

Find .

4 (a) Explain method of finding the distribution and density function for a sum of statistically independent random

variables. (b)

Find constant b:

Is a valid joint density function. 5 (a) Two Gaussian random variable and have variance 2 2 = 4 respectively and correlation co-

efficient rotation by an angle results in new random variable Y1 and Y2 are uncorrelated what is (b) Prove the mean value of weighted sum of random variables equal to the weighted sum of mean sum of

mean value. 6 (a) A random process is defined by where A is a continuous random variable uniformly distribution on

(0, 1) determine if it is wide sense stationary. (b) Explain the classification of random process with neat sketch. 7 (a) Write and explain the properties of auto correlation wide sense stationary random process. (b) If the random process X(t) had no periodic components and if X(t) is non-zero mean then:

Lim

8 (a) For a random process x(t) derive expression for power density spectrum. (b) A random noise X(t) having power spectrum is applied to a network for which h(t) =

The network response is denoted by Y(t): (i) Find the average power of X (t). (ii) Find the power spectrum of Y(t). (iii) Find the average power of Y (t).

*****

4

*****

Code: 9A04303

B.TECH II Year I Semester (R09) Regular & Supplementary Examinations November 2012 PROBABILITY THEORY & STOCHASTIC PROCESSES

(Common to Electronics & Instrumentation Engineering, Electronics & Control Engineering, and Electronics & Communication Engineering)

Time: 3 hours Max. Marks: 70 Answer any FIVE questions

All questions carry equal marks

*****

1. (a) Explain the following: (i) Random experiment (ii) Trial (iii) Event (iv) sample space. (b) Find the probability of obtaining 14 with 3 dice using Baye’s theorem.

2. (a) Explain with an example discrete, continuous and mixed random variables. (b) Explain CDF with its properties.

3. What is a characteristic function? Explain its properties with its proofs.

4. (a) State and explain ‘Central Limit Theorem’. (b) The joint pdf of two random variable ‘X’ and ‘Y’ is given by

fXY (x, y) = K, (x2 + 2y);0 otherwise

x = 0,1,2, y = 1,2,3,4

Find (i) The ‘K’ value (ii) P(X = 1, Y = 2) (iii) P(X ≤ 1, Y ≥ 3). (iv) fX(x)& fY (y) (v) fY

X y1 & fX

Y x2 .

5. (a) Explain about joint moments about the origin with an example. (b) ‘X’ is a random variable with mean ‘4’ and variance ‘3’. Another random variable ‘Y’ is related

to ‘X’ as Y=2X+7. Determine (i) E[X2] (ii) E[Y] (iii) var [Y] (iv) RXY

6. (a) Differentiate WSS & SSS. (b) Prove the following: (i) |RXX (τ)| ≤ RXX (0) .(ii) RXX (−τ) = RXX (τ) . (iii) RXX (0) = E[X2(t)].

7. (a) What is meant by co-variance and explain its properties. (b) A random process X(t) = Acosω0t + Bsinω0t, where ω0 is constant and A & B are random

variables. If A and B are uncorrelated zero mean having same variance σ2 but different density functions then show that X(t) is a wide sense stationary.

8. (a) Give the relation between cross power spectrum and cross correlation function. (b) A random process has a power spectrum

SXX (ω) = 4 − ω2

9 , |ω| ≤ 6

0 elsewhere

Find (i) . Average power (ii) RMS bandwidth.

*****

1

Code: 9A04303

B.TECH II Year I Semester (R09) Regular & Supplementary Examinations November 2012 PROBABILITY THEORY & STOCHASTIC PROCESSES

(Common to Electronics & Instrumentation Engineering, Electronics & Control Engineering, and Electronics & Communication Engineering)

Time: 3 hours Max. Marks: 70 Answer any FIVE questions

All questions carry equal marks

*****

1. (a) Explain the following: (i) Principles of counting (ii) probability as a relative frequency. (b) How many positive integers less than 1000 have no common factor with 1000?

2. (a) Define a random variables and give the conditions for a function to be a random variable. (b) Explain about normal distribution with its properties.

3. Find the mean and variance of Binomial Distribution and Poisson Distribution.

4. (a) Explain about joint density function with its properties. (b) The joint density function of two random variables ‘X’ ad ‘Y’ is given by

fXY (x, y) = 1π√3

e−23 (x2−xy +y2) . Determine the marginal probability density function fX(x) and

fY (y).

5. (a) Explain about jointly Gaussian Random variables. (b) A random variable ‘Z’ has pdf fZ(z) = ae−a(z−b)u(z− b). Show that the characteristic function of

z is ∅z(ω) = aa−jw

e−jwb has the probability function P(x) = 132x, x = 1,2,3−−− N.

6. (a) What is random process and classify it and explain. (b) A stationary continuous random process ‘X’ is differentiable and X(t) is its derivates. Show that

EX(t) = 0

7. (a) A WSS noise process N(t) has ACF RNN (τ) = Pe−3|τ| . Find PSD and plot both ACF and PSD. (b) If X(t) is WSS, find RYY (τ) and hence SYY (ω) in terms of SXX (ω) for the product device shown

in below fig. x(t)⎯

SXX (ω)product

y(t)⎯

SYY (ω)

↑Acos ωt

8. If a random process X(t) = A0(cosω0t + θ), where A0 & ω0 are constants and ‘θ′ is a uniformly distributed random variable in the interval (0,π) Find

(i) Whether X(t) is WSS process? (ii) Power in X(t) by time averaging of its second moment. (iii) The power spectral density ofX(t).

*****

2

Code: 9A04303

B.TECH II Year I Semester (R09) Regular & Supplementary Examinations November 2012 PROBABILITY THEORY & STOCHASTIC PROCESSES

(Common to Electronics & Instrumentation Engineering, Electronics & Control Engineering, and Electronics & Communication Engineering)

Time: 3 hours Max. Marks: 70 Answer any FIVE questions

All questions carry equal marks

***** 1. (a) Explain the following: (i) probability (ii) Axioms. (b) From the urn containing ‘n’ balls any numbers of balls are drawn. Show that the probability of

drawing an even number of ball is (2n−1 − 1)(2n − 1)

2. (a) Differentiate pmf and pdf.

(b) Show that fX(x) = 1σ√2π

e(x−μ )2

2σ2 , −α0 < 𝑥𝑥 < α0,σ > 0 is a distribution function.

3. (a) The pdf of the random variable ‘X’ follows

fX(x) =1

2θe−|x−θ|

θ ,−α0 < 𝑥𝑥 < α0 Find m.g.f . Hence or otherwise find E(X) and var(X).

4. Joint pdf fXY (x, y) of two continuous random variables ′X′and ′Y′ is given by

fXY (x, y) = K, e−(2x+y) for x, y ≥ 0;0 otherwise

Where K is constant. (i) find K value and fX(x) and fY (y). (ii) Are ′X′and ′Y′ are statistically independent? (iii) Determine joint CDF and marginal distribution function. (iv) Determine the conditional density functions.

5. (a) Explain about joint central moments with an example. (b) A random variable ′Z′ with pdf fZ(z) = 1

2;−1 ≤ Z ≤ 1. Another random variable random

variables (RV)s ′X′ = Z and ′Y′ = Z2 . Show that ′X′and ′Y′ are uncorrelated.

6. (a) What is meant by stochastic process and classify with an example to each. (b) Check the following for WSS. (i) RXX (t, t + τ) = cosst e−|t+τ|) (ii) RXX (t, t + τ) = sin2τ/(1 + τ2)

(iii) RXX (t, t + τ) = −10−|τ| (iv) RX(t, t + τ) = 5e−|τ|

7. (a) Explain in detail the cross power spectral density. (b) If X(t) and Y(t) are random processes. Prove that

(i). SXY (ω) = SYX (−ω) = SYX∗ (ω) (ii) SXY (ω) = SYX (ω) if X(t) & Y(t) are uncorrelated WSS

random processes.

8. (a) Give the relation between power spectrum and auto correlation function. (b) Find the cross correlation function corresponding to the cross power spectrum

SXY = 6/[(9 + ω2)(3 + jω)2]

*****

3

Code: 9A04303

B.TECH II Year I Semester (R09) Regular & Supplementary Examinations November 2012 PROBABILITY THEORY & STOCHASTIC PROCESSES

(Common to Electronics & Instrumentation Engineering, Electronics & Control Engineering, and Electronics & Communication Engineering)

Time: 3 hours Max. Marks: 70 Answer any FIVE questions

All questions carry equal marks

***** 1. (a) State the Baye’s theorem and prove it. (b) There are 300 students in a class room. It is known that 180 can program ‘JAVA’, 120 in ‘C++’

30 in ‘SQL’, 12 in ‘JAVA’ and ‘SQL’, 18 in ‘C++’ and ‘SQL’, 12 in JAVA and ‘C++’ and 6 in all three languages. (i) A student is selected at random. What is the probability that she can program in exactly two languages. (ii) Two students are selected at random. What is the probability that they can (a) Both program in ‘JAVA’ (b) Both program only in ‘JAVA’.

2. (a) Explain about pdf. (b) The diameter of a cable ′X′ is taken to be a random variable with pdf fX(x) = 6x(1 − x), 0 ≤ x ≤

1 (i) verify fX(x) is a pdf or not.(ii) Determine ′b′ such that P(x < 𝑏𝑏) = P(x > 𝑏𝑏).

3. (a) Define and explain moments of a random variable. (b) Find the moment generating function of a random variable.

4. (a) Find the density of W=X+Y , where the densities of ‘X’ and ‘Y’ to be fX(x) = u(x) −u(x − 1)& fY (y) = u(y) − u(y − 1).

(b) Explain the following (i) Joint distribution function. (ii) conditional distribution function (iii) Marginal distribution function.

5. (a) Show that the variance of variance of a weighted sum of uncorrelated random variables equals

the weighted sum of t he variances of the random variables. (b) Explain about transformation of multiple random variables.

6. Explain about the concept stationarity in detail connected with stochastic processes.

7. (a) Explain about WSS and prove any two properties of it. (b) Y(t) = X(t) cos(ω0t + θ), where X(t) is a random process and ′θ′ is uniformly distributed over

the interval (0,2π). Determine under what conditions is Y(t) wide seuce stationary. Assume ′θ′ and ′X(t) are statistically independent and ω0 is constant.

8. (a) Write different types of band pass processes with band limited processes. (b) Find the rms band width of the power spectrum

SXX (ω) = Acosπω2W

, |ω| ≤ W

0, |ω| ≤ W

Where ω > 0 & 𝑊𝑊 > 0 are constants?

*****

4

Code: 13A04304

B.Tech II Year I Semester (R13) Regular Examinations December 2014PROBABILITY THEORY & STOCHASTIC PROCESSES

(Electronics and Communication Engineering) Time: 3 hours Max. Marks: 70

PART – A (Compulsory Question)

***** 1 Answer the following: (10 X 02 = 20 Marks)

(a) State Baye’s theorem.(b) Three coins are tossed in succession. Find out the probabilities of occurrence of two consecutive

heads.(c) State central limit theorem.(d) Find the expected value of the face value while rolling fair die?(e) Define cross-covariance function.(f) Give any two examples for poisson random process.(g) A random process has the power density spectrum SXX (ω) = . Find the average power in the

process.(h) What is power spectral density? Mention its importance.(i) Define the following random process: (i) Band limited. (ii) Narrow band.(j) What are the two conditions that are to be satisfied by the power spectrum to be a valid

power density spectrum?

PART – B (Answer all five units, 5 X 10 = 50 Marks)

UNIT - I

2 (a) A pack contains 4 white and 2 green pencils, another contains 3 white and 5 green pencils. If one pencil is drawn from each pack, find the probability that (i) Both are white. (ii) One is white and another is green

(b) Explain about joint and conditional probability. OR

3 (a) Consider the experiment of tossing four fair coins. The random variable X is associated with the number of tails showing. Compute and sketch the CDF of X.

(b) Define probability density function. List its properties.

UNIT - II

4 (a) Let X and Y be jointly continuous random variables with joint density function

fXY(x,y) = xy ; for x>0, y>0 = 0; otherwise

(i) Check whether x and y are independent.(ii) Find P (x≤1, y≤1).

(b) How expectation is calculated for two random variables? OR

5 (a) Prove the following: Var (ax+by) = a2 var(x) + b2 var(y) + 2ab cov(x,y)

(b) Explain central limit theorem.

Contd. in page 2

Page 1 of 2

R13

Code: 13A04304

UNIT - III

6 (a) Explain about mean-ergodic process.(b) If x (t) is a stationary random process having mean = 3 and auto correlation function:

RXX (τ) = 9 + 2 . Find the mean and variance of the random variable. OR

7 (a) Explain the significance of auto correlation.(b) Find auto correlation function of a random process whose power spectral density is given by

UNIT – IV

8 (a) Briefly explain the concept of cross power density spectrum.(b) Find the cross correlation of functions sin ωt and cos ωt.

OR 9 (a) The power spectral density of a stationary random process is given by

SXX (ω) = A; -k < ω < k = 0; otherwise

Find the auto correlation function. (b) Discuss the properties of power spectral density.

UNIT – V

10 (a) A Gaussian random process X (t) is applied to a stable linear filter. Show that the random process Y(t) developed at the output of the filter is also Gaussian.

(b) Discuss about cross correlation between the input X (t) and output Y (t). OR

11 (a) Derive the relation between PSDs of input and output random process of an LTI system. (b) The input voltage to an RLC series circuit is a stationary random process X(t) with E[X (t)] = 2 and

RXX (τ) = 4 + exp (-2 ). Let Y (t) is the voltage across capacitor. Find E[Y(t)] .

*****

Page 2 of 2

R13

Code: 9A04303 B.Tech II Year I Semester (R09) Regular & Supplementary Examinations December/January 2013/14

PROBABILITY THEORY AND STOCHASTIC PROCESSES (Common to EIE, E.Con.E & ECE)

Time: 3 hours Max Marks: 70 Answer any FIVE questions

All questions carry equal marks *****

*****

1 (a) Define the following with example: (i) Sample space. (ii) Event. (iii) Mutually exclusive event. (iv) Independent event. (v) Exhaustive event.

(b) When a die is tossed find the probabilities of the event A = odd number shown up; B = Number larger than 3 shown up then find out A B and A B.

2 (a) Define probability density function and explain with an example and write its properties. (b) The random variable has following density function:

(i) Find value of k. (ii) P ( . 1 2 3 4 5 6 7 k 2k 2k 3k

3 (a)

The density function of a random variable X is:

(i) E [x], (ii) E [(x-1)2], (iii) E [3x -1]. (b) For Poisson distributions find out moment generating function and characteristic function. 4 (a) Distinguish between joint distribution and marginal distribution. (b)

Joint probability density function of two random variables X and Y.

Find: (i) Value of ‘a’; (ii) . 5 (a) Explain relation between marginal and joint characteristic function. (b) In a control system a random voltage is known to have mean value and a second

moment If the voltage x is amplified by an amplifier that gives an output 2, 2 and

6 (a) State the conditions for wide sense stationary random process. (b) Explain the classifications of random process. 7 (a) What is an ergodic random process, present the necessary expression to support the argument? (b) Consider a random process where is a random variable

uniformly distributed over where is any real number find 8 (a) Explain the relation between power spectrum and auto correlation function of random process. (b) Write any two properties of cross power density spectrum.

1

Code: 9A04303 B.Tech II Year I Semester (R09) Regular & Supplementary Examinations December/January 2013/14

PROBABILITY THEORY AND STOCHASTIC PROCESSES (Common to EIE, E.Con.E & ECE)

Time: 3 hours Max Marks: 70 Answer any FIVE questions

All questions carry equal marks *****

1 (a) State and prove Baye’s theorem of probability. (b) Two similar boxes ‘A’ and ‘B’ contain 2 white and 3 red balls 4 white and 5 red balls respectively. If

a ball is selected at random from one of the boxes, then find the probability that the box is ‘B’ when the ball is red.

2 (a) Write the method for defining conditional event. (b)

Check whether following is a probability density function or not

3 (a) Explain variance and skew. (b) The mean and variance of binomial distribution are 4 and 4/3 respectively find (c) Find the expected value of the number on a die when thrown.

4 (a) Define and explain the conditional properties. (b)

The joint probability density function of two random variables x and y given by

(i) Find the value of c. (ii) Marginal distribution function X and Y.

5 (a) Write the expression for expected value of a function of random variables and prove that the mean

value of a weighted sum of random variables equals the weighted sum of mean values. (b) Two Gaussian random variables X1 and X2 defined by the mean and co-variance matrices

two new random variable s Y1 and Y2 are formed using the transformation

. Find the matrices , and also find the torrelation co-efficient of and i.e., .

6 (a) Explain the types of statistical averages. (b) Define a random process where ‘A’ is Gaussian random variable with zero mean

and variance 2 is X(t) is stationary in any sense. 7 (a) Define cross correlation and write its properties. (b) A random process is defined as where is a uniform random variable over

(0, 2 Verify the process is ergodic in the mean sense and auto correlation sense. 8 (a) Write the relation between cross power spectrum and cross correlation function. (b) If power spectral density of a random process is given by

Find the auto correlation function.

*****

2

Code: 9A04303 B.Tech II Year I Semester (R09) Regular & Supplementary Examinations December/January 2013/14

PROBABILITY THEORY AND STOCHASTIC PROCESSES (Common to EIE, E.Con.E & ECE)

Time: 3 hours Max Marks: 70 Answer any FIVE questions

All questions carry equal marks *****

Contd. in Page 2

Page 1 of 2

1 (a) What is sample space? Explain the discrete and continuous sample space. (b) In a box there are 100 resistors having resistances and tolerances as shown in the table.

Resistance Tolerance

Total 5% 10%

22 10 14 24 47 28 16 44 100 24 8 32 Total 62 38 100

Let a resistor be selected from the box and assume each resistor are equally likely occur. Define three events ‘A’ as draw a 47 Ω resistor, ‘B’ as draw a resistor with 5% tolerance and ‘C’ as draw a 100 Ω resistor. (i) The probability of drawing a 47 Ω resistor given that the resistor drawn is 5%. (ii) The probability of drawing a 47 Ω resistor given that the resistor drawn is 100 Ω. (iii) The probability of drawing a resistor of 5% tolerance given resistor is 100 Ω. (iv) Find remaining conditional probabilities.

2 (a) Sketch probability density function and probability distribution function of: (i) Exponential distribution. (ii) Rayleigh distribution. (iii) Uniform distribution.

(b) Define conditional distribution function and write their properties. 3 (a) State and prove any three properties of variance of a random variable. (b) For binomial density prove:

(i) . (ii) . (iii) 2 = npq. 4 (a) Define and explain joint distribution function and joint density function of two random variables X

and Y. (b) The joint space for two random variable X and Y and its corresponding probabilities are shown in

table 1, 1 2, 2 3, 3 4, 4

0.2 0.3 0.35 0.15 Find and plot (a). Marginal distribution of X and Y.

(c) (d) .

3

Code: 9A04303

*****

Page 2 of 2

5 (a) Show that the variance of a weighted sum of uncorrelated random variables equals the weighted sum of the variance of the random variables.

(b)

Two random variables X and Y have the density function:

(i) Find all the order moment. (ii) Find covariance. (iii) X and Y and uncorrelated.

6 (a) Explain the concept of random process. (b) Distinguish between:

(i) Deterministic random process and non-deterministic random process. (ii) Stationary and non- stationary random process.

7 (a) S.T Where A is normal random variable with zero mean and unity

variance and uniformly distributed is Assume A and are independent random variable.

(b) Discuss Gaussian random process and state its properties. 8 (a) State and explain four properties of power density spectrum of a random process. (b) If find the spectral density function, where a and b are constant.

3

Code: 9A04303 B.Tech II Year I Semester (R09) Regular & Supplementary Examinations December/January 2013/14

PROBABILITY THEORY AND STOCHASTIC PROCESSES (Common to EIE, E.Con.E & ECE)

Time: 3 hours Max Marks: 70 Answer any FIVE questions

All questions carry equal marks

1 (a) Give the classical and axiomatic definitions of probability. (b) A card is drawn from a well shuffled pack of playing cards. What is the probability that in either a spade or

ace? (c) What is the probability: (i) Leap year selected at random will contain 53 Sundays?

(ii) Non leap year selected at random will contain 53 Sundays? 2 (a) Define random variable and explain types of random variable. (b) In an experiment of rolling a die and flipping a coin the random variable ( is chosen such that.

(i) A coin tail (H) outcome corresponds to positive value of that are equal in magnitude to twice the number that shown on die. Map the elements of random variable into points on the real line and explain.

3 (a) Write about transformation. (b) Let x be a random variable defined by density function:

Find .

4 (a) Explain method of finding the distribution and density function for a sum of statistically independent random

variables. (b)

Find constant b:

Is a valid joint density function. 5 (a) Two Gaussian random variable and have variance 2 2 = 4 respectively and correlation co-

efficient rotation by an angle results in new random variable Y1 and Y2 are uncorrelated what is (b) Prove the mean value of weighted sum of random variables equal to the weighted sum of mean sum of

mean value. 6 (a) A random process is defined by where A is a continuous random variable uniformly distribution on

(0, 1) determine if it is wide sense stationary. (b) Explain the classification of random process with neat sketch. 7 (a) Write and explain the properties of auto correlation wide sense stationary random process. (b) If the random process X(t) had no periodic components and if X(t) is non-zero mean then:

Lim

8 (a) For a random process x(t) derive expression for power density spectrum. (b) A random noise X(t) having power spectrum is applied to a network for which h(t) =

The network response is denoted by Y(t): (i) Find the average power of X (t). (ii) Find the power spectrum of Y(t). (iii) Find the average power of Y (t).

*****

4

*****

Code: 9A04303

B.TECH II Year I Semester (R09) Regular & Supplementary Examinations November 2012 PROBABILITY THEORY & STOCHASTIC PROCESSES

(Common to Electronics & Instrumentation Engineering, Electronics & Control Engineering, and Electronics & Communication Engineering)

Time: 3 hours Max. Marks: 70 Answer any FIVE questions

All questions carry equal marks

*****

1. (a) Explain the following: (i) Random experiment (ii) Trial (iii) Event (iv) sample space. (b) Find the probability of obtaining 14 with 3 dice using Baye’s theorem.

2. (a) Explain with an example discrete, continuous and mixed random variables. (b) Explain CDF with its properties.

3. What is a characteristic function? Explain its properties with its proofs.

4. (a) State and explain ‘Central Limit Theorem’. (b) The joint pdf of two random variable ‘X’ and ‘Y’ is given by

fXY (x, y) = K, (x2 + 2y);0 otherwise

x = 0,1,2, y = 1,2,3,4

Find (i) The ‘K’ value (ii) P(X = 1, Y = 2) (iii) P(X ≤ 1, Y ≥ 3). (iv) fX(x)& fY (y) (v) fY

X y1 & fX

Y x2 .

5. (a) Explain about joint moments about the origin with an example. (b) ‘X’ is a random variable with mean ‘4’ and variance ‘3’. Another random variable ‘Y’ is related

to ‘X’ as Y=2X+7. Determine (i) E[X2] (ii) E[Y] (iii) var [Y] (iv) RXY

6. (a) Differentiate WSS & SSS. (b) Prove the following: (i) |RXX (τ)| ≤ RXX (0) .(ii) RXX (−τ) = RXX (τ) . (iii) RXX (0) = E[X2(t)].

7. (a) What is meant by co-variance and explain its properties. (b) A random process X(t) = Acosω0t + Bsinω0t, where ω0 is constant and A & B are random

variables. If A and B are uncorrelated zero mean having same variance σ2 but different density functions then show that X(t) is a wide sense stationary.

8. (a) Give the relation between cross power spectrum and cross correlation function. (b) A random process has a power spectrum

SXX (ω) = 4 − ω2

9 , |ω| ≤ 6

0 elsewhere

Find (i) . Average power (ii) RMS bandwidth.

*****

1

Code: 9A04303

B.TECH II Year I Semester (R09) Regular & Supplementary Examinations November 2012 PROBABILITY THEORY & STOCHASTIC PROCESSES

(Common to Electronics & Instrumentation Engineering, Electronics & Control Engineering, and Electronics & Communication Engineering)

Time: 3 hours Max. Marks: 70 Answer any FIVE questions

All questions carry equal marks

*****

1. (a) Explain the following: (i) Principles of counting (ii) probability as a relative frequency. (b) How many positive integers less than 1000 have no common factor with 1000?

2. (a) Define a random variables and give the conditions for a function to be a random variable. (b) Explain about normal distribution with its properties.

3. Find the mean and variance of Binomial Distribution and Poisson Distribution.

4. (a) Explain about joint density function with its properties. (b) The joint density function of two random variables ‘X’ ad ‘Y’ is given by

fXY (x, y) = 1π√3

e−23 (x2−xy +y2) . Determine the marginal probability density function fX(x) and

fY (y).

5. (a) Explain about jointly Gaussian Random variables. (b) A random variable ‘Z’ has pdf fZ(z) = ae−a(z−b)u(z− b). Show that the characteristic function of

z is ∅z(ω) = aa−jw

e−jwb has the probability function P(x) = 132x, x = 1,2,3−−− N.

6. (a) What is random process and classify it and explain. (b) A stationary continuous random process ‘X’ is differentiable and X(t) is its derivates. Show that

EX(t) = 0

7. (a) A WSS noise process N(t) has ACF RNN (τ) = Pe−3|τ| . Find PSD and plot both ACF and PSD. (b) If X(t) is WSS, find RYY (τ) and hence SYY (ω) in terms of SXX (ω) for the product device shown

in below fig. x(t)⎯

SXX (ω)product

y(t)⎯

SYY (ω)

↑Acos ωt

8. If a random process X(t) = A0(cosω0t + θ), where A0 & ω0 are constants and ‘θ′ is a uniformly distributed random variable in the interval (0,π) Find

(i) Whether X(t) is WSS process? (ii) Power in X(t) by time averaging of its second moment. (iii) The power spectral density ofX(t).

*****

2

Code: 9A04303

B.TECH II Year I Semester (R09) Regular & Supplementary Examinations November 2012 PROBABILITY THEORY & STOCHASTIC PROCESSES

(Common to Electronics & Instrumentation Engineering, Electronics & Control Engineering, and Electronics & Communication Engineering)

Time: 3 hours Max. Marks: 70 Answer any FIVE questions

All questions carry equal marks

***** 1. (a) Explain the following: (i) probability (ii) Axioms. (b) From the urn containing ‘n’ balls any numbers of balls are drawn. Show that the probability of

drawing an even number of ball is (2n−1 − 1)(2n − 1)

2. (a) Differentiate pmf and pdf.

(b) Show that fX(x) = 1σ√2π

e(x−μ )2

2σ2 , −α0 < 𝑥𝑥 < α0,σ > 0 is a distribution function.

3. (a) The pdf of the random variable ‘X’ follows

fX(x) =1

2θe−|x−θ|

θ ,−α0 < 𝑥𝑥 < α0 Find m.g.f . Hence or otherwise find E(X) and var(X).

4. Joint pdf fXY (x, y) of two continuous random variables ′X′and ′Y′ is given by

fXY (x, y) = K, e−(2x+y) for x, y ≥ 0;0 otherwise

Where K is constant. (i) find K value and fX(x) and fY (y). (ii) Are ′X′and ′Y′ are statistically independent? (iii) Determine joint CDF and marginal distribution function. (iv) Determine the conditional density functions.

5. (a) Explain about joint central moments with an example. (b) A random variable ′Z′ with pdf fZ(z) = 1

2;−1 ≤ Z ≤ 1. Another random variable random

variables (RV)s ′X′ = Z and ′Y′ = Z2 . Show that ′X′and ′Y′ are uncorrelated.

6. (a) What is meant by stochastic process and classify with an example to each. (b) Check the following for WSS. (i) RXX (t, t + τ) = cosst e−|t+τ|) (ii) RXX (t, t + τ) = sin2τ/(1 + τ2)

(iii) RXX (t, t + τ) = −10−|τ| (iv) RX(t, t + τ) = 5e−|τ|

7. (a) Explain in detail the cross power spectral density. (b) If X(t) and Y(t) are random processes. Prove that

(i). SXY (ω) = SYX (−ω) = SYX∗ (ω) (ii) SXY (ω) = SYX (ω) if X(t) & Y(t) are uncorrelated WSS

random processes.

8. (a) Give the relation between power spectrum and auto correlation function. (b) Find the cross correlation function corresponding to the cross power spectrum

SXY = 6/[(9 + ω2)(3 + jω)2]

*****

3

Code: 9A04303

B.TECH II Year I Semester (R09) Regular & Supplementary Examinations November 2012 PROBABILITY THEORY & STOCHASTIC PROCESSES

(Common to Electronics & Instrumentation Engineering, Electronics & Control Engineering, and Electronics & Communication Engineering)

Time: 3 hours Max. Marks: 70 Answer any FIVE questions

All questions carry equal marks

***** 1. (a) State the Baye’s theorem and prove it. (b) There are 300 students in a class room. It is known that 180 can program ‘JAVA’, 120 in ‘C++’

30 in ‘SQL’, 12 in ‘JAVA’ and ‘SQL’, 18 in ‘C++’ and ‘SQL’, 12 in JAVA and ‘C++’ and 6 in all three languages. (i) A student is selected at random. What is the probability that she can program in exactly two languages. (ii) Two students are selected at random. What is the probability that they can (a) Both program in ‘JAVA’ (b) Both program only in ‘JAVA’.

2. (a) Explain about pdf. (b) The diameter of a cable ′X′ is taken to be a random variable with pdf fX(x) = 6x(1 − x), 0 ≤ x ≤

1 (i) verify fX(x) is a pdf or not.(ii) Determine ′b′ such that P(x < 𝑏𝑏) = P(x > 𝑏𝑏).

3. (a) Define and explain moments of a random variable. (b) Find the moment generating function of a random variable.

4. (a) Find the density of W=X+Y , where the densities of ‘X’ and ‘Y’ to be fX(x) = u(x) −u(x − 1)& fY (y) = u(y) − u(y − 1).

(b) Explain the following (i) Joint distribution function. (ii) conditional distribution function (iii) Marginal distribution function.

5. (a) Show that the variance of variance of a weighted sum of uncorrelated random variables equals

the weighted sum of t he variances of the random variables. (b) Explain about transformation of multiple random variables.

6. Explain about the concept stationarity in detail connected with stochastic processes.

7. (a) Explain about WSS and prove any two properties of it. (b) Y(t) = X(t) cos(ω0t + θ), where X(t) is a random process and ′θ′ is uniformly distributed over

the interval (0,2π). Determine under what conditions is Y(t) wide seuce stationary. Assume ′θ′ and ′X(t) are statistically independent and ω0 is constant.

8. (a) Write different types of band pass processes with band limited processes. (b) Find the rms band width of the power spectrum

SXX (ω) = Acosπω2W

, |ω| ≤ W

0, |ω| ≤ W

Where ω > 0 & 𝑊𝑊 > 0 are constants?

*****

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