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Code No: X0421

II B.Tech I Semester, Supplementary Examinations, May – 2012

PROBABILITY THEORY AND STOCHASTIC PROCESSES

(Com. to ECE, ECC)

Time: 3 hours Max Marks: 80

Answer any FIVE Questions

All Questions carry Equal Marks

~~~~~~~~~~~~~~~~~~~~~~~~~

1. a) A survey of 100 companies shows that 75 of them have installed wireless local area network

(WLANS) on their premises. If three of these companies are chosen at random without

replacement, what is the probability that each of the three has installed WLANS.

b) State and prove the Baye’s theorem. (8M+8M)

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

≤≤−

<<

=

otherwise

z x x

x x

x f X

0

12

10

i) Find the CDF of X

ii) Find P [0.2 < X <0.8]

iii) Find P [0.6 < X <1.2]

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

3. a) The PDF of a random variable X is given by ( ) ( ) .3081 / 942

≤≤−= x x x x f X Find the

mean, variance and third moment of X.

b) The random variable X has the following PDF ( )

<<−

=

Otherwise

x x f X

0

213

1

If we desire Y = 2X +3, what is the PDF of Y. (8M+8M)

4. a) Two random variables X and Y have zero mean and variance 162= X σ and 36

2=Y σ , if

their correlation coefficient is 0.5 determine the following

i) The variance of the sum of X and Yii) The variance of the difference of X and Y

b) State and prove the ‘Central Limit Theorem’ (8M+8M)

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Code No: X0421

5. a) A joint probability density function is ( )

elsewhere

y

x

y x f XY 40

60

024

1

, <<

<<

=

Find out the expected value of the function ( ) ( )2

, XY Y X g = b) Prove that the moment generating function of the sum of two independent variables is the

product of their moment generating function (8M+8M)

6. a) Explain the following

i) Stationarity

ii) Ergodicity

iii) Statistical independence with respect to random processes

b) State and prove the properties of cross correlation function (8M+8M)

7. a) A random process Y(t) has the power spectral density ( )64

92+

=ω

ω YY S

Find i) The average power of the process

ii) The Auto correlation function

b) State the properties of power spectral density (8M+8M)

8. Write short notes on the following

i) Thermal noise

ii) Effective noise temperature (8M+8M)

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~~~~~~~~~~~~~~~~~~~~~~~~~

1. a) Explain the following

i) Probability as a relative frequency

ii) Joint Probability

iii) Total Probability

iv) Independent Events (2M+3M+3M+2M)

b) Tom is planning to pick up a friend at the airport. He has figured out that the plane is late

80% of the time when it rains, but only 30% of the time when it does not rain. If the weatherforecast that morning calls for a 40% chance of rain, what is the probability that the plane will

be late? (6M)

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

( ) ( )

>

≤<−+

−≤

=

21

22sin1

20

π

π π

π

x

x xK

x

xF X

Find i) the value of Kii) the PDF of X.

b) Write short notes on ‘Rayleigh Distribution’ (8M+8M)

3. a) If X has the probability density function

( ) ( )

otherwise. 0

0xx-exp

=

>= x f X

Find the expected value of g(x) = exp (3X/5).

b) Show that any characteristic function ( ) ( ) ( ) 10satisfies =≤ X X X φ ω φ ω φ

(8M+8M)

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Code No: X0421

4. a) The joint density function of two continuous random variables X and Y is given by

( )( )

22

x e

2

1,

22

2 σ π

y y x f XY

+= Determine the marginal density function

b) The Joint probability density function of two random variables X and Y is given by

( )

otherwise. 0

4y0yx0 xy,

=

≤≤≤≤= β y x f XY

i) Determine the value of β

ii) Are X and Y are statistically independent (8M+8M)

5. a) An exponential random variable Z has a probability density function

( ) ( )b zub za

ea z Z

f −−−

=

Show that the characteristic function of Z, ( ) b j-e

ω

ω ω φ

ja

a

Z −=

b) X is random variable with mean value x and variance2

X σ Y is another random variable

given by Y= ax+b where a and b are constant. Show that X and Y are orthogonal if

( )( ) X E

X E ab

2

−=

(8M+8M)

6. a) Explain the concept of Random Process

b) Explain Autocorrelation function and state its properties

c) Write short notes on “Poisson Random Process” (4M+6M+6M)

7. a) Find the Auto Correlation function of the random process whose power spectral density is

( )W

W PS

XX >

≤

=

ω

ω πω ω

0

2W

cos

b) Prove that the power spectrum and Autocorrelation function of the random process form a

Fourier transform pair (8M+8M)

8. Explain the followingi) Properties of Band Limited Processes

ii) Average Noise figure

iii) Resistive Noise (6M+4M+6M)

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~~~~~~~~~~~~~~~~~~~~~~~~~

1. a) Five men out of 100 and 25 women out of 100 are colour blind. A clour blind person is

chosen at random. What is the probability that the person is a male? Assume that males and

females are in equal proportion

b) Distinguish between the following terms

i) Discrete and continuous sample spaces

ii) Joint Probability

iii) Conditional Probabilityiv) Axioms of Probability (8M+8M)

2. a) A random variable X has the Pdf

( )otherwise

x

0x1

K

xf 2X

α α <<−

+=

Determine the value of K and find the CDF FX(x). Also evaluate P(X≥0).

b) Write short note on ‘Poisson distribution’. (8M+8M)

3. a) If the continuous random variable X has the Pdf ( ) ( )

otherwise 0

2x1- 1x 9

2

xf X

<<+=

Find the Pdf of Y = X2.

b) A random variable X has the density ( ) ( )

otherwise

x

0

9x3 85x- 23

3

xf 2

X

≤≤−+=

Find the following moments i) m0 ii)m1 (8M+8M)

4. a) State the properties of Joint Probability density function

b) If X and Y are independent random variables with density function

( ) ( ) ( ) ( ) yu y xue x f x

X

-3y

Y e3f and ==− Find the density function of Z = X+Y (8M+8M)

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5. a) For any two random variables X and Y, Show that [ ]( ) [ ]( ) [ ]( )2

122

122

12

Y E X E Y X E +≤+

b) The random variable X is uniformly distributed between 1 and 2. If S is the sum of 40

independent experimental values of X, evaluate P(55< S≤65) using Central Limit Theorem.

(8M+8M)

6. a) The random process X(t) is given by X(t) = Y cos (2πt) t ≥ 0 where Y is random variable

that is uniformly distributed between 0 and 2 Find the expected value and autocorrelation

function of X(t)

b) Write short notes on “Gaussian Random Processes” (8M+8M)

7. a) A random process X(t) has PSD given by ( )elsewhere 0

6 9

4wS

2

XX

≤−=

ω ω

Find i) The average powerii) Auto correlation function

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

8. a) A wide sense stationary process X (t) is the input to a linear system whose impulse response

is h (t) = 2 e-7t, t≥0. If the auto correlation function of the process is ( )τ

τ 4−

= e XX

R and the

output process is Y(t) , find the following

i) The PSD of Y(t)

ii) The cross spectral density SXY( )

iii) The cross correlation function ( )τ XY

R

b) Write short notes on the following

i) Available power gain

ii) Equivalent noise Band Width (8M+8M)

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Code No: X0421



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~~~~~~~~~~~~~~~~~~~~~~~~~

1. a) Define and Explain the following with examples

i) Discrete sample space

ii) Continuous sample space

iii) Conditional probability

iv) Total probability

b) A box of 30 diodes is known to contain 5 defective ones. If two diodes are selected atrandom without replacement, what is the probability that atleast one of these diodes is defective

(8M+8M)

2. a) A random variable has a PDF

( ) .5525 2

<<−

−

= x x

c x f

X Find

i) The value of C

ii) P(X>2)

iii) P(X<3)iv) P(X<3/X>2)

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

3. a) A random variable X has characteristic function ( )1 0

1 1

>

≤−=

ω

ω ω φ w X

Find the density function

b) Find the moment generating function about origin of the Poisson distribution (8M+8M)

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4. a) The Joint Pdf of X and Y is given by

( ) [ ] α α π

<<−++= y x y xy y x f XY ,05.1 / 2x-exp428.1

1, 22

i) Find the Pdfs for X and Y

ii) Are X and Y independent

b) X is a random variable uniformly distributed between 0 and 5, Y is another random variable,

independent of X, uniformly distributed between +3 and -3. The new random variable is given

by W= X+Y. What is the Pdf of new random variable ? (8M+8M)

5. a) For two zero Gaussian random variables X and Y show that their Joint characteristic

function is ( ) [ ].22

1-exp,2

2

2

21

2

1

2

21 wY y x x XY σ ω ω σ σ ρ ω σ ω ω φ ++=

b) Write short notes on “Jointly Gaussian random variable” (8M+8M)

6. a) Explain the following write to random process

i) Strict sense stationary

ii) Mean ergodic process

b) Distinguish between autocorrelation and cross correlation and state the properties of auto

correlation function (8M+8M)

7. a) For a random process X(f) assume that ( ) ( ).τ τ K

X eK R−

= Show that the power spectral

density is given by ( ) 2

1

2

+

=

K

S XX

ω ω

b) Write short notes on “Cross power density spectrum” (8M+8M)

8. a)Write short notes on “Frequency response of LTI system”

b) Explain the following

i) Effective noise temperature

ii) Average noise figure (8M+8M)

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