05210401 P R O B A B I L I T Y T H E O R Y A N D S T O C H A S T I C P R O C E S S

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Code No: R05210401
Set No. 1
II B.Tech I Semester Supplimentary Examinations, November 2008
PROBABILITY THEORY AND STOCHASTIC PROCESS
( Common to Electronics & Communication Engineering, Electronics &
Telematics and Electronics & Computer Engineering)
Time: 3 hours Max Marks: 80
Answer any FIVE Questions
All Questions carry equal marks

1. (a) What is an event and explain discrete and continuous events with an example.
(b) Discuss joint and conational probability.
(c) Determine the probability of a card being either red or a queen. [6+6+4]

2. (a) De ne cumulative probability distribution function. And discuss distribution
functions speci c properties.
(b) What are the conditions for the function to be a random variable? Discuss.
What do you mean by continuous and discrete random variable? [8+8]

3. (a) A discrete random variable X is an outcome of throwing a fair die. Find mean
and variance of X.
(b) Find the skew and coe cient of skewness for a Rayleigh random variable X
-(x-2)/10 2
5
[6+10]
X ( ) = (x-2) 02
4. Discrete random variables X and Y have a joint distribution function
XY ( ) = 0 1 ( + 4) ( - 1) + 0 15 ( + 3) ( + 5) + 0 17 ( + 1) ( - 3)+
0 05 ( ) ( - 1) + 0 18 ( - 2) ( + 2) + 0 23 ( - 3) ( - 4)+
0 12 ( - 4) ( + 3)
Find

(a) Sketch X Y ( )
(b) marginal distribution functions of X and Y.

(c) P(-1 X = 4 -3 Y = 3) and
(d) Find P(X = 1 Y = 2). [4+6+3+3]

5. (a) Random variables X and Y have the joint density f XY (x,y)= 1/24
0 x b and 0 y 4 what is the expected value of the function g(x,y) =
( )2
(b) Show that the correlation coe cient satis es the expression | | = | 11| ( 11 20) =
1. [8+8]

6. Let X(t) be a stationary continuous random pro cess that is di erentiable. Denote
its time derivative by X(t)

(a) Show that • × ( ) = 0.

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Code No: R05210401
Set No. 1
(b) Find × × ( ) in terms of ×× ( )sss [8+8]

7. (a) Determine which of the following functions are valid PSDS.

i. 2
6 3 2 +3
ii. exp[-( - 1)2]
iii. 2 4 +1 - ( )
iv. 4 1+ 2 +j 6 .
[4 ×3 = 12]
(b) De ne RMS bandwidth of PSD and explain. [4]

8. (a) For the network shown in gure 8 nd the transfer function of the system. [8]
(b) De ne the following systems
i. LTI system
ii. Causal system
iii. stable system
iv. Noise Bandwidth. [4×2=8]

Figure 8

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Code No: R05210401
Set No. 2

1. (a) Is probability relative frequency of occurrence of some event? Explain with
an example.
(b) De ne an event and explain discrete and continuous event with an example.
(c) One card is drawn from a regular deck of 52 cards. What is the probability of
the card being either red or a king? [6+6+4]

2. (a) De ne cumulative probability distribution function. And discuss distribution
functions speci c properties.
(b) What are the conditions for the function to be a random variable? Discuss.
What do you mean by continuous and discrete random variable? [8+8]

3. (a) The density function of a random variable X is ( ) = 5 -5x 0 = = 8
0
Find
i. E[X],
ii. E[( - 1)2]
iii. E[3X-1]
(b) If the mean and variance of the binomial distribution are 6 and 1.5 respectively.
Find E[X - P(X = 3)] [8+8]

4. If the joint PDF of (X,Y) is given by X Y ( ) = -(x2 +y2 ) = 0 = 0
0
Find

(a) constant ‘C’ so that this is a valid joint density function.
(b) marginal distribution functions of X and Y.

(c) show that X and Y are independent random variables.
(d) Find P(X = 1 Y = 1). [4+6+3+3]

5. (a) Show that the variance of a weighted sum of uncorrected random variables
equals the weighted sum of the variances of the random variables.
(b) Two random variables X and Y have joint characteristic function
X Y( 1 2) = exp(-2 2 1-8 2 2)
i. Show that X and Y are zero mean random variables.

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Code No: R05210401
Set No. 2
ii. are X and Y are correlated. [8+8]

6. Let X(t) be a stationary continuous random pro cess that is di erentiable. Denote
its time derivative by X(t)

(a) Show that • × ( ) = 0.

(b) Find × × ( ) in terms of ×× ( )sss [8+8]

7. (a) A WSS noise pro cess N(t) has ACF NN ( ) = -3|t |. Find PSD and plot
both ACF and PSD
(b) Find Y Y ( ) and hence Y Y ( ) interns of X X ( ) for the product device
shown in gure 7 if X(t)is WSS. [8+8]

Figure 7
8. (a) For the network shown in gure 8 nd the transfer function of the system. [8]
(b) De ne the following systems
i. LTI system
ii. Causal system
iii. stable system
iv. Noise Bandwidth. [4×2=8]

Figure 8

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Code No: R05210401
Set No. 3

1. (a) De ne probability with an Axiomatic Approach.
(b) In a box there are 100 resistos having resistance and tolerance as shown in
table 1. If a resistor is chosen with same likeliho od of being chosen for the
three events, A as “draw a 470 resistor”, B as “draw a 100 resistor”,
determine joint probabilities and conditional probabilities.
Table 1

Number of resistors in a box having given resistance and tolerance. [6+10]

Resistance ( ) Tolerance
5% 10% Total
22 10 14 24
47 28 16 44
100 24 8 32
Total 62 38 100

2. (a) De ne conditional density function of a random variable and discuss their
properties.
(b) A random variable X has density function f(x) =1/4 for -2 x 2 and 0
elsewhere. Obtain
i. P(x 1)
ii. P( |x| 1). [10+6]

3. (a) De ne and explain moments of a random variable.
(b) A random variable has the probability function ( ) = 1 32 x = 1 2 3 .
Find the moment generating function. [8+8]

4. (a) State and prove central limit theorem.
(b) Find the density of W = X + Y where the densities of X and Y are assumed
to be
X ( ) == [ ( ) - ( - 1)] Y ( ) = [ ( ) - ( - 1)], [8+8]

5. (a) Let X be a random variables Another random variable Y related to X as. Y
= aX+b where a and b are constants.
i. Find the covariance of X and Y.

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Code No: R05210401
Set No. 3
ii. Find the correlation coe cient of X and Y.
(b) Let X and Y be de ned by X= Cos & Y= Sin where is a random variable
uniformly distributed over (0 2 ).
i. Show that X and Y are uncorrelated
ii. Show that X and Y are not independent. [8+8]

6. ¯X = 6 and RXX(t t+ ) = 36 + 25 exp (- | | ) for a random process X(t). In-
dicate which of the following statements are true based on what is known with
certainty X(t)

(a) is rst order stationary
(b) has total average power of 61W

(c) is ergo dic
(d) is wide sense stationary
(e) has a perio dic component
(f ) has an ac power of 36w. [16]

7. (a) Find the ACF of the following PSD’s

i. ( ) = 157+12 2
(16+ 2 )(9+ 2 )
ii. ( ) = 8
(9+ 2 )2
(b) State and Prove wiener-Khinchin relations. [8+8]

8. (a) A Signal x(t) = u(t) exp (- t ) is applied to a network having an impulse
response h(t)= u(t) exp (- t). Here & are real positive constants.
Find the network response? (6M)
(b) Two systems have transfer functions 1( ) & 2( ). Show the transfer
function H( ) of the cascade of the two is H( ) = 1( ) 2 ( ).
(c) For cascade of N systems with transfer functions n( ) , n=1,2,.... .N show
that H( ) = n( ). [6+6+4]

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Code No: R05210401
Set No. 4

1. (a) What is ‘ total probability’ and show that total probability P(A) is
P(A) = P(A n S) = P[UNn = 1(A n Bn)] =N n=1P(A n Bn) with Venn dia-
gram.
(b) In a box there are 100 resistors having resistance and tolerance as shown in
table. Let a resistor be selected from the box and assume each resistor has
the same likelihood of being chosen. For the three events; A as “draw a 47
resistor,” B as “draw a resistor with 5% tolerance” and C as “draw a 100
resistor” calculate the joint probabilities. [10+6]

Table 1
Numbers of resistors in a box having given resistance and tolerance.

Resistance( ) Tolerance
5% 10% Total
22 10 14 24
47 28 16 44
100 24 8 32
Total 62 38 100

2. (a) De ne and explain the following density functions
i. Binomial
ii. Exponential
iii. Uniform
iv. Rayleigh.
(b) What is density function of a random variable x, if x is guassian. [12+4]

3. (a) State and prove properties of variance of a random variable
(b) Let X be a random variable de ned by the density function
16 cos( 8) -4 = = 4 Find E[3X] and E[ 2]. [8+8]
X()=p 0

4. If the joint PDF of (X,Y) is given by Find x,y( ) = ( ) ( )[1 - -x/2 - -y/2 +
-(x+y)/2] Find

(a) Plot , x,y( )

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Code No: R05210401
Set No. 4
(b) Find P(0 5 X 1 5)
(c) P(X = 1 Y = 2) and
(d) P(-0 5 X = 2 1 Y = 3) . [4+4+4+4]

5. (a) let Y = 1 + 2 + ............+ N be the sum of N statistically independent
random variables i, i=1,2.............. N. If Xi are identically distributed then
nd density of Y, y(y).

(b) Consider random variables 1 and 2 related to arbitrary random variables X
and Y by the coordinate rotation. 1=X Cos + Y Sin 2 = -X Sin + Y
Cos

i. Find the covariance of 1 and 2, CY1Y2
ii. For what value of , the random variables 1 and 2 uncorrelated. [8+8]

6. A random process X(t) has periodic sample functions as shown in gure6, where
B,T and 4to = T are constants but is a random variable uniformly distributed
on the interval (0,T)

(a) nd the rst order distribution function of X(t)
(b) nd the rst order density function

(c) nd E[X(t)], E[ 2(t)] and 2. [4+4+8]

Figure 6

7. (a) A WSS noise pro cess N(t) has ACF NN ( ) = -3|t |. Find PSD and plot
both ACF and PSD
(b) Find Y Y ( ) and hence Y Y ( ) interns of X X ( ) for the product device
shown in gure 7 if X(t)is WSS. [8+8]

Figure 7

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Code No: R05210401
Set No. 4
8. (a) A Signal x(t) = u(t) exp (- t ) is applied to a network having an impulse
response h(t)= u(t) exp (- t). Here & are real positive constants.
Find the network response? (6M)
(b) Two systems have transfer functions 1( ) & 2( ). Show the transfer
function H( ) of the cascade of the two is H( ) = 1( ) 2 ( ).
(c) For cascade of N systems with transfer functions n( ) , n=1,2,.... .N show
that H( ) = n( ). [6+6+4]

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05210401 P R O B A B I L I T Y T H E O R Y A N D S T O C H A S T I C P R O C E S S

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