Jawaharlal Nehru Technological University Kakinada 2007 B.Tech Computer Science and Engineering PROBABILITY THEORY AND STOCHASTIC PROCESS (2) - Question Paper

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Code No: R059210401 Set No. 3
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: three hours Max Marks: 80
ans any 5 ques.
All ques. carry equal marks
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1. (a) describe probability based on set theory and fundamental axioms.
(b) When 2 dice are thrown, obtain the probability of getting the sums of 10 or
11. [8+8]
2. (a) describe cumulative probability distribution function. And explain 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 presumed 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) discuss the concept of a transformation of a random variable X
(b) A Gaussian random variable X having a mean value of zero and variance 1 is
transformed to an a different random variable Y by a square legal regulations transformation.
obtain 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) obtain 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
obtain 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 ? + Y Sin ? Y2 = -X Sin ? + Y
Cos ?
i. obtain the covariance of Y1 and Y2, CY1Y2
ii. For what value of ?, the random variables Y1 and Y2 uncorrelated. [8+8]
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6. (a) describe cross correlation function of 2 random processes X(t) and Y(t) and
state the properties of cross correlation function.
(b) let 2 random processes X(t) and Y(t) be described by
X(t) = A cos ?0t + B sin ?0t
Y(t) = B cos ?0t - A sin ?0t
Where A and B are random variables and ?0 is a constant. presume A and
B are uncorrelated, zero mean random variables with identical variance. obtain
the cross correlation function RXY (t,t+t ) 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(? ). obtain the PSD of dx(t)
dt
(b) Prove that Sxx (? ) = Sxx (-? )
(c) If R(t ) = ae|by|. obtain 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 t = 2e-4|_| is applied to the network shown in figure 8a obtain
i. SXX (? )
ii. IH(? )I2
iii. SY Y (? ). [4+4+2]
Figure 8a
(b) Write short notes on various kinds of noises. [6]
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Earning:   Approval pending.