PROBABILITY THEORY AND STOCHASTIC PROCESS JNTU UNIVERSITY PREVIOUS QUESTION PAPER MODEL QUESTION PAPER

PROBABILITY THEORY AND STOCHASTIC PROCESS JNTU UNIVERSITY PREVIOUS QUESTION PAPER MODEL QUESTION PAPER

PROBABILITY THEORY AND STOCHASTIC PROCESS JNTU UNIVERSITY PREVIOUS QUESTION PAPER MODEL QUESTION PAPER

II B.Tech I Semester Regular Examinations, November 2007

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) Discuss Joint and conditional probability.
(b) When are two events said to be mutually exclusive? Explain with an example?
(c) Determine the probability of the card being either red or a king when one card
is drawn from a regular deck of 52 cards. [6+6+4]
2. (a) Define rayleigh density and distribution function and explain them with their
plots.
(b) Define and explain the guassian random variable in brief?
(c) Determine whether the following is a valid distrbution function. F(x) = 1-
exp(-x/2) for x ) 0 and 0 elsewhere. [5+5+6]
3. (a) State and prove properties of characteristic function of a random variable X
(b) Let X be a random variable defined by the density function
fX(x) = 5


4 (1 − x4) 0 < x 1
0 elsewhere
. Find E[X] ,E[X2] and variance. [8+8]
4. The joint space for two random variables X and Y and corresponding probabilities
are shown in table
Find and Plot
(a) FXY (x, y)
(b) marginal distribution functions of X and Y.
(c) Find P(0.5 < X < 1.5),
(d) Find P(X 1,Y 2) and
(e) Find P(1 < X 2,Y 3).
X, Y 1,1 2,2 3,3 4,4
P 0.05 0.35 0.45 0.15
[3+4+3+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
φX, Y(ω1,ω2) = exp(−2ω2
1−8ω2
2).
i. Show that X and Y are zero mean random variables.
ii. are X and Y are correlated. [8+8]
6. Let X(t) be a stationary continuous random process that is differentiable. Denote
its time derivative by ˙ X(t).
(a) Show that E h •
× (t)i = 0.
(b) Find R× ˙× (τ ) in terms of R×× (τ )sss [8+8]
7. (a) Derive the expression for PSD and ACF of band pass white noise and plot
them
(b) Define various types of noise and explain. [8+8]
8. (a) Define the following random processes
i. Band Pass
ii. Band limited
iii. Narrow band. [3×2 = 6]
(b) A Random process X(t) is applied to a network with impulse response h(t) =
u(t) exp (-bt)
where b > 0 is ω constant. The Cross correlation of X(t) with the output Y
(t) is known to have the same form:
RXY (τ ) = u(τ )τ exp (-bY)
i. Find the Auto correlation of Y(t)
ii. What is the average power in Y(t). [6+4]

II B.Tech I Semester Regular Examinations, November 2007

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) With an example define and explain the following:
i. Equality likely events
ii. Exhaustive events.
iii. Mutually exclusive events.
(b) In an experiment of picking up a resistor with same likelihood of being picked
up for the events; A as “draw a 47 
 resistor”, B as “draw a resistor with
5% tolerance” and C as “draw a 100 
 resistor” from a box containing 100
resistors having resistance and tolerance as shown below. Determine joint
probabilities and conditional probabilities. [6+10]
Table 1
Number of resistor 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) What is binomial density function? Find the equation for binomial distrbution
function.
(b) What do you mean by continuous and discrete random variable? Discuss the
condition for a function to be a random variable. [6+10]
3. (a) Define moment generating function.
(b) State properties of moment generating function.
(c) Find the moment generating function about origin of the Poisson distribution.
[3+4+9]
4. (a) Define conditional distribution and density function of two random variables
X and Y
(b) The joint probability density function of two random variables X and Y is
given by
f(x, y) = a(2x + y2) 0 x 2 , 2 y 4
0 elsewhere
. Find
1 of 2
Code No: R059210401 Set No. 2
i. value of ‘a’
ii. P(X 1,Y > 3). [8+8]
5. (a) let Xi, i = 1,2,3,4 be four zero mean Gaussian random variables. Use the joint
characteristic function to show that E {X1 X2 X3 X4} = E[X1 X2] E[X3 X4]
+ E[X1X3]E[X2X4] + E[X2X3] E[X1X4]
(b) Show that two random variables X1 and X2 with joint pdf.
fX1X2(X1,X2) = 1/16 |X1|< 4, 2 < X2< 4 are independent and orthogonal.[8+8]
6. A random process Y(t) = X(t)- X(t +τ ) is defined in terms of a process X(t) that
is at least wide sense stationary.
(a) Show that mean value of Y(t) is 0 even if X(t) has a non Zero mean value.
(b) Show that σY2= 2[RXX(0) − RXX(τ )]
(c) If Y(t) = X(t) +X(t + τ ) find E[Y(t)] and σY 2. [5+5+6]
7. (a) If the PSD of X(t) is Sxx(ω ). Find the PSD of dx(t)
dt
(b) Prove that Sxx (ω ) = Sxx (-ω )
(c) If R(τ ) = ae|by|. Find the spectral density function, where a and b are con-
stants. [5+5+6]
8. (a) A Signal x(t) = u(t) exp (-αt ) is applied to a network having an impulse
response h(t)= ω u(t) exp (-ω t). Here α & ω are real positive constants.
Find the network response? (6M)
(b) Two systems have transfer functions H1( ω) & H2( ω). Show the transfer
function H(ω) of the cascade of the two is H(ω ) =H1( ω) H2 (ω ).
(c) For cascade of N systems with transfer functions Hn(ω) , n=1,2,.... .N show
that H( ω) = πHn(ω). [6+6+4]

II B.Tech I Semester Regular Examinations, November 2007

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) Define probability based on set theory and fundamental axioms.
(b) When two dice are thrown, find the probability of getting the sums of 10 or
11. [8+8]
2. (a) Define cumulative probability distribution function. And discuss distribution
function specific properties.
(b) The random variable X has the discrete variable in the set {−1,−0.5, 0.7, 1.5, 3}
the corresponding probabilities are assumed to be {0.1, 0.2, 0.1, 0.4, 0.2}. plot
its distribution function and state is it a discrete or continuous ditribution
function. [8+8]
3. (a) Explain the concept of a transformation of a random variable X
(b) A Gaussian random variable X having a mean value of zero and variance one is
transformed to an another random variable Y by a square law transformation.
Find the density function of Y. [8+8]
4. Discrete random variables X and Y have a joint distribution function
FXY (x, y) = 0.1u(x + 4)u(y − 1) + 0.15u(x + 3)u(y + 5) + 0.17u(x + 1)u(y − 3)+
0.05u(x)u(y − 1) + 0.18u(x − 2)u(y + 2) + 0.23u(x − 3)u(y − 4)+
0.12u(x − 4)u(y + 3)
Find
(a) Sketch FXY (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) let Y = X1 + X2 + ............+XN be the sum of N statistically independent
random variables Xi, i=1,2.............. N. If Xi are identically distributed then
find density of Y, fy(y).
(b) Consider random variables Y1 and Y2 related to arbitrary random variables X
and Y by the coordinate rotation. Y1=X Cos θ + Y Sin θ Y2 = -X Sin θ + Y
Cos θ
i. Find the covariance of Y1 and Y2, CY1Y2
ii. For what value of θ, the random variables Y1 and Y2 uncorrelated. [8+8]
1 of 2
Code No: R059210401 Set No. 3
6. (a) Define cross correlation function of two random processes X(t) and Y(t) and
state the properties of cross correlation function.
(b) let two random processes X(t) and Y(t) be defined by
X(t) = A cos ω0t + B sin ω0t
Y(t) = B cos ω0t - A sin ω0t
Where A and B are random variables and ω0 is a constant. Assume A and
B are uncorrelated, zero mean random variables with same variance. Find
the cross correlation function RXY (t,t+τ ) and show that X(t) and Y(t) are
jointly wide sense stationary. [6+10]
7. (a) If the PSD of X(t) is Sxx(ω ). Find the PSD of dx(t)
dt
(b) Prove that Sxx (ω ) = Sxx (-ω )
(c) If R(τ ) = ae|by|. Find the spectral density function, where a and b are con-
stants. [5+5+6]
8. (a) A Stationary random process X(t) having an Auto Correlation function
RXX τ = 2e−4| | is applied to the network shown in figure 8a find
i. SXX (ω )
ii. IH(ω )I2
iii. SY Y (ω ). [4+4+2]
Figure 8a
(b) Write short notes on different types of noises. [6]

II B.Tech I Semester Regular Examinations, November 2007

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) Define and explain the following with an example:

i. Equally likely events
ii. Exhaustive events
iii. Mutually exclusive events
(b) Give the classical definition of probability.
(c) Find the probability of three half-rupee coins falling all heads up when tossed
simultaneously. [6+4+6]
2. (a) What is poisson random variable? Explain in brief.
(b) What is binomial density and distrbution function?
(c) Assume automobile arrives at a gasoline station are poisson and occur at an
average rate of 50/hr. The station has only one gasoline pump. If all cars are
assumed to require one minute to obtain fuel. What is the probability that a
waiting line will occur at the pump? [5+5+6]
3. (a) Define moment generating function.
(b) State properties of moment generating function.
(c) Find the moment generating function about origin of the Poisson distribution.
[3+4+9]
4. Given the function f(x, y) = (x2 + y2)/8π x2 + y2 < b
0 elsewhere
(a) Find the constant ‘b’ so that this is a valid joint density function.
(b) Find P(0.5b < X2 + Y2 < 0.8b). [7+9]
5. Three statistically independent random variables X1,X2 and X3 have mean values
¯X
1= 3, ¯X2= 6 and ¯X3= −2. Find the mean values of the following functions.
(a) g(X1,X2,X3) = X1 + 3X2 + 4X3
(b) g(X1,X2,X3) = X1 X2 X3
(c) g(X1,X2,X3) = −2X1,X2 −3X1 X3 + 4X2 X3
(d) g (X1,X2,X3) = X1+X2+X3. [16]


6. Statistically independent zero mean random processes X(t) and Y(t) have auto
correlations functions
RXY (τ ) = e - | | and
RYY(τ ) = cos (2 τ ) respectively.
(a) find the auto correlation function of the sum W1(t) = X(t) + Y(t)
(b) find the auto correlation function of difference W2(t) = X(t) - Y(t)
(c) Find the cross correlation function of W1(t) and W2(t). [5+5+6]
7. (a) Find the ACF of the following PSD’s
i. S (ω) = 157+12!2
(16+!2)(9+!2)
ii. S (ω) = 8
(9+!2)2
(b) State and Prove wiener-Khinchin relations. [8+8]
8. A random noise X(t) having power spectrum SXX(ω) = 3
49+!2 is applied to a to a
network for which h(t) = u(t)t2 exp(−7t). The network response is denoted by Y(t)
(a) What is the average power is X(t)
(b) Find the power spectrum of Y(t)
(c) Find average power of Y(t). [5+6+5]