r07(nov,10,reg)k

7/27/2019 R07(NOV,10,REG)K

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Code No: X0421/R07

II B.Tech I Semester (R07) Regular Examinations, Nov- 2010

PROBABILITY THEORY AND STOCHASTIC PROCESSES

(Com. to ECE, ETM, ECC)

Time: 3 hours Max Marks: 80Answer any FIVE Questions

All Questions carry equal marks

1. a) Define probability as a relative frequency and state the axioms of probability (4M)

b) State and prove the Bayes theorem (6M)

c) A letter is known as to have come from either TATANAGAR or CALCUTTA. On the

envelope, just two consecutive lasers TA are visible. Find the probability that the letter hascome from CALCUTTA. (6M)

2. a) Distinguish between Distribution and Density function of random variable state the

properties of density function (8M)

b) Write short notes on “Gaussian distribution” (8M)

3. a) The characteristic function for a Gaussian random variable x, having a mean value of 0, is

)(ω φ ×

= exp

)2 / 22

ω σ ×

−

Find all the moments of × using )(φ ×

(8M)

b) A random variable θ is uniformly distributed in the interval ( )21,θ θ where 1θ and 2θ are

real and satisfy π θ θ <<≤ 210 . Find and sketch the probability density function of the

transferred random variable θ cos=Y (8M)

4. a) State and prove the center limit theorem (8M)

b) Joint distribution of X and Y is given by .0,0,)()(

,

22

≥≥=+−

y xe xy B y x f y x

Find whether X and Y are independent. Find the conditional density of X given Y= y

(8M)

5. a) Find the mean value of the function22

),( Y X Y X g += ,where X and Y are random

variables defined by the density function =)( , y x f XY 2

2 / )(

2

222

σ π

σ

ey x +−

with2σ a constant (8M)

b) Two random variables X and Y have Joint characteristic function

( ).82exp),(2

2

2

1

2

21 ω ω ω ω φ −−= XY Show that X and Y are both zero mean random variables

and they are uncorrelated (8M)

6. a) Explain the following concepts in brief.

i) Random processes.

ii) Stationarity and statistical independence (8M)b) Distinguish between Autocorrelation and cross correlation and state the properties of

cross correlation function. (8M)

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Code No: X0421/R07

7. a) Determine which of the following functions can and cannot be valid power density

spectrum. For those that are not, explain why.

i) ])1([exp2

−− ω ii)62

1

4

ω ω ω

j++

iii) )(1

4

2

ω δ ω

ω −

+iv)

21

)3(cos

ω + (8M)

b) State the properties of cross power density spectrum (8M)

8. a) Explain the following

i) Effective Noise temperatureii)Resistive Noise (8M)

b) Consider a linear system as shown in figure below

x(t) is the input and y(t) is the output of the system.

The auto correlation of x(t) is Rxx )(τ = 5 )(τ δ . Find the PSD, autocorrelation function and

mean square value of the output y(t).

(8M)

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ω j+6

1

x(t) y(tͿ

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1. a) Explain the following

i) Probability

ii) Conditional probability

iii) Independent events

iv) Discrete sample space (8M)

b) State and prove the Bayes theorem (8M)

2. a) A random variable X has the distribution function

)(650

)(212

1

n xun

xF n

X −=¦=

Find the probabtres P { }5.5≤×<∞−

P { }6>×

and P { }85 ≤×< (8M)

b) Write short notes on “Poisson distribution” (8M)

3. a) The pdf of random variable X is given as¯®-

=

,0

,3507.0)(

x x f X

otherwise

x 30 <<

Find thei) Mean

ii) Mean of the square

iii) Variance of the random variable X (8M)

b) Show that the characteristic function of a random variable having the binomial density is N

je p p

X »¼º

«¬ª

+−=ω

ω φ 1)((8M)

4. a) Distinguish between Joint density function and Marginal density function. State theproperties of Joint density function. (8M)

b) Statistically independent random variables x and y have probability densities

)4(32

3)( 2

x x f X −= , 22 ≤≤− x

= 0, elsewhere

[ ])1()1(2

1)( −−+= yu yu y f Y

Find the exact probability density of the sum W= X+Y (8M)

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Code No: X0421/R07

5. a) Write short notes on “Jointly Gaussian Random variable’s (8M)

b) A Joint density function is given as¯®- +

=,0

),5.2()( ,

y x y x f XY

elsewhere

y x 20,20 <<<<

Find all the Joint moments mnk, n and k=0,1,…. (8M)

6. a)Explain the following with respect to Random process

i) Correlation erogodic processes ii) Wide sense stationarity

(8M)

b) State the properties of

i) Auto correlation function ii) Cross correlation function (8M)

7. a) Prove that power density spectrum )( xxS and time average of process Auto correlation

function ª º)(τ XX R A for wide cense stationary process form a Fourier transform pair i.e.,

),()(τ xx XX S R ↔(8M)

b) A random process has the power density spectrum

42

2

)1(

6)(

ω

ω ω

+= xxS . Find the average power in the process. (8M)

8. a) Write short notes on the following

i) Average Noise figure

ii) Thermal noise (8M)

b) For the following receiver, calculate the effective noise temperature.

.aG = 26dB Ga=17dB

Te = 40k. F = 6dB

ȕ = 25 MHZ (8M)

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Antenna

Maser TWL=0.4 dB


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Code No: X0421/R07






1. a) Define the following

i) Joint probability

ii) Conditional probability

iii) Total probability

iv) Independent Event (8M)b) A missile can be accidently launched in two relays A and B both have failed. The

Probabilities of A and B failing are known to be 0.01 and 0.03 respectively. It is also known

that B is more likely to fail (probability 0.06) if A has failed

i) What is the probability of an accidental missile launch?

ii) What is the probability that A will fail if B has failed?

iii) Are the Events ‘A’ fails and ‘B’ fails statistically independent? (8M)

2. a) The pdf of a random variable X is given by

b xaK x f X ≤≤=)(

= 0 , otherwise where K is a constant

i) Determine the value of K

ii) let a=1 and b=2 calculate p )( c x ≤ for C=0.5. (8M)

b) Write short notes on “Rayleigh distribution and density function” (8M)

3. a) Find the mean value of Random variable X, if X has the following density function

,0

,1)(2 x x f −

= whereelse

x 11 ≤≤−

(8M)

b) A random variable X is uniformly distributed in (0, 6). If X is transformed to a near

random variable Y = 2( X -3)2-4, find

i) The density of Y, ii) Y and iii)2

yσ (8M)

4. a) Define Joint distribution function and state its properties (8M)

b) Statically independent random variables X and Y have respective densities)5(exp)(5)( x xu x f

X −=

)2(exp)(2)( y yu y f Y −= , find the density of the sum W= X+Y. (8M)

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Code No: X0421/R07

5. a) Random variables X and Y have Joint density function

33 ,11for,40 / )()( 2

, <<−<<−+= y x y x y x f XY

= 0, else where

i) Find all the second order moments of X and Yii) What are the variances of X and Y

iii) What is the correlation coefficient (8M)

b) For N random variables, show that

1)0,0(1

)1

(1

=≤+ "

!

"

!

N X X n

N X X

φ ω ω φ (8M)

6. a) Explain stationary and ergodic Random Processer (8M)

b) Prove that [ ])0()0(2

1)( YY XX XX R R R +≤τ

(8M)

7. a) Find the power spectrum for the random process X(t) with Autocorrelation function

shown in figure below

(8M)

b) Discuss the properties of power spectral density (8M)

8. a) Prove that the output power spectral density equals the input power spectral density

multiplied by the squared magnitude of the transform of the filter. (8M)

b) Write short notes on “Band limited and narrow band processes” (8M)

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Rxx (τ

A

-T T τ

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Code No: X0421/R07




Time: 3 hours Max Marks: 80

Answer any FIVE QuestionsAll Questions carry equal marks

1. a) In a bolt factory, machines A, B, C manufacture 30%, 30%, 40% of total output

respectively. If the total of their outputs, 4, 5, 3 percents are defective bolts. A bolt is drawn

at random from the product and is found to be defective. What are the probabilities that it

was manufactured by machines A, B and C. (8M)

b) Distinguish between Joint probability, conditional probability and total Probability (8M)

2. a) Let X be a continuous random variable with distribution function

k x

x f +=6

)( , 30 ≤≤ x

= 0, other wise

i) Find the value of k

ii) Find P ª º21 ≤×≤(8M)

b) Write short notes on “poission distribution function” (8M)

3. a) If X is a normal random variable, find the mean and variance of this random variable.(8M)

b) A random variable X undergoes the transformation Y= X

a, where ‘a’ is the real number.

Find the density function of Y. (8M)

4. a) The Joint Probability density function of two dimensional random variable(X,Y) is given

by xy y x f 9

8

),( = , 21 ≤≤≤ y x

= 0, elsewhere

Find the marginal density functions of X and Y (8M)

b) State and power the central limit theorem (8M)

5. a) Random variable X and Y have the Joint density

°̄

°®-

=

036

1

),( y x f XY ,

elsewhere

yand x 6080 <<<<

Find the Expected value of the function 2)(),( Y X Y X g = (8M)

b) A Joint density is given by°¯

°®

-<<<<+

= otherwise

3y0and2x0 ,

0

)5.0(43

2

),(

2 y x y x f XY

i) Find all the first and second order moment

ii) Find the Covarianceiii) Are X and Y uncorrelated (8M)

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Code No: X0421/R07

6. a) Given the random process x(t)=A cos )( 0 t + B Sin )( 0 t ω where0

ω is a constant and A

and B are uncorrelated zero mean random variables having different density functions but

the same variances 2σ show that X(t) wide sense stationary but not strictly stationary.

(8M)b) State the properties of Auto correlation and cross correlation (8M)

7. a) If the Autocorrelation function of a loss process is RX (τ ) = K)(τ K

e−

show that its

spectral density is given by2

1

2)(

¸ ¹

·¨©

§ +

=

K

S XX ω

ω

(8M)

b) Prove that cross power spectrum and cross correlation function form Fourier transform

pair (8M)

8. a) A signal )(exp)()( t t ut x α −= is applied to a network having an impulse response

),(exp)()(2

t W W t ut h −+= where 00 >> W and α are real constants. Find the

Network’s response y(t) (8M) b) Show that the effective noise temperature of ‘n’ networks in cascade is given by

nggg

neT

gg

eT

g

eT eT

eT

....2

,1

........

21

3

1

21

++++=

(8M)

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