Model Question paper
Programme
Course
Title
Faculty
Time
: B.Tech (ECE) & B.Tech (ECM)
Probability Theory and Random Processes
:
Semester : WIN 2020-21
Code
: ECE2005
Max.
Marks
:
: Dr. K. Mohanaprasad
: Three Hours
100
Answer all questions
Q.No.
Sub.
Sec.
Question Description
Marks
Given that the function
1
[10]
𝑏(𝑥 + 𝑦)2
− 2 < 𝑥 < 2 𝑎𝑛𝑑 − 3 < 𝑦 < 3
𝑓𝑋,𝑌 = {
}
0
𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
Find the constant b such that this is a valid joint density function and also determine the marginal
density function 𝑓𝑥 (𝑥) 𝑎𝑛𝑑 𝑓𝑦 (𝑦).
2.
Two statistically independent random variables X1 and X2 have the same probability density given by [5]
2𝑥𝑖
0 ≤ 𝑥𝑖 < 𝑎
𝑓𝑋 (𝑥) = {
𝑎2
𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒 𝑖𝑛 𝑥𝑖
0
For i=1 and 2, where a > 0 is a constant.
Find the exact density of the sum 𝑊 = 𝑋1 + 𝑋2 and also compute mean and variance of W using
central limit theorem.
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3.
[10]
Two random variables X and Y are defined by 𝑋̅ = 0, 𝑌̅ = −1, ̅̅̅̅
𝑋 2 = 2 , ̅𝑌̅̅2̅ = 4 𝑎𝑛𝑑 𝑅𝑋𝑌 =
2
2
2
̅̅̅̅̅
̅
̅
̅
̅
̅, 𝑈
̅, 𝑊 , 𝑌 , 𝜎𝑋 ,
−2. Two new random variable W and U are 𝑊 = 2𝑋 + 𝑌, 𝑈 = −𝑋 − 3𝑌. Find 𝑊
2
𝜎𝑌 𝑎𝑛𝑑 𝑅𝑊𝑈 .
4.
Statistically independent random variable X and Y have moments 𝑚10 = 2, 𝑚20 = 14, 𝑚02 = [5]
12, 𝑎𝑛𝑑 𝑚11 = −6. Find the moment 𝜇22 .
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5
6.
2
2
Gaussian random variables 𝑋1 𝑎𝑛𝑑 𝑋2 , for which ̅̅̅
𝑋1 = 2, 𝜎𝑋1
= 9, ̅̅̅
𝑋2 = −1, 𝜎𝑋2
= 4 and 𝐶𝑋1,𝑋2 = [5]
−3, are transformed to new random variables 𝑌1 𝑎𝑛𝑑 𝑌2 according to 𝑌1 = −𝑋1 + 𝑋2 , 𝑌2 = −2𝑋1 −
2
2
3𝑋2 . Find 𝑋̅12 , ̅𝑋22 , 𝜌𝑋1,𝑋2 , 𝜎̅𝑌1
, 𝜎̅𝑌2
, 𝐶𝑌1,𝑌2 .
A Number of practical systems have “square-law” detectors that produce an output W(t) that is the [10]
square of its input Y(t). Let the detector’s output be defined by W (t )  Y 2 (t )  X 2 (t ) Cos 2 (0 t   )
where 0 is a constant. X(t) is second-order stationary and  is a random variable independent of
X(t) and uniform on (0, 2π). Find E[W(t)], Rww (t,t+τ) and check whether or not W(t) is wide-sense
stationary.
7
Statistically independent, zero-mean random process X(t) and Y(t) have autocorrelation functions [5]

RXX ( )  e  and RXX ( )  cos(2) respectively.
i.
Find the auto correlation function of the sum W1(t) = X(t) +Y(t).
ii.
Find the auto correlation function of the difference W2(t) = X(t) -Y(t).
iii.
Find the cross correlation function of W1(t) and W2(t).
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8
9
A stationary random sequence 𝑥(𝑛) with mean µx=4 and auto covariance 𝐶𝑋 (𝑘) = [10]
|𝑘| ≤ 3
4 − |𝑘|,
{
} is applied as input to a linear shift invariant system who impulse response
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
ℎ(𝑛) 𝑖𝑠 ℎ(𝑛) = 𝑢(𝑛) − 𝑢(𝑛 − 4) where 𝑢(𝑛) is a step sequence. The output of this system is
another random sequence 𝑦(𝑛). Determine the mean sequence 𝜇𝑦 , the cross covariance 𝐶𝑋𝑌 (𝑘) and
the auto covariance 𝐶𝑌𝑌 (𝑘)𝑜𝑓 𝑦(𝑛).
A causal LTI system, which is described by the difference equation
[10]
1
1
𝑦(𝑛) = 𝑦(𝑛 − 1) + 𝑥(𝑛) + 𝑥(𝑛 − 1) is driven by a zero mean WSS process with autocorrelation
2
3
𝑅𝑋𝑋 (𝑙) = 0.5|𝑙|
Determine the Power Spectral Density and the Cross Power Spectral Density.
10
A Random process X (t ) is applied to a network with impulse response h(t )  u (t ) t exp( bt )
where b > 0 is a constant. The cross-correlation of X (t ) with the output Y(t) is known to have the
[10]
same form: RXY ( )  u( )  exp( b ) . Find the autocorrelation of Y (t ) and the average power in
Y (t ) .
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11
Periodic samples of the autocorrelation function of white noise N(t) with period Ts are defined by
2
𝑘=0
𝑅𝑁𝑁 (𝑘𝑇𝑠 ) = {𝜎𝑁
}
0
𝑘≠0
A random sequence Y[n] is formed by adding the white noise sequence N[n] to a one-unit delayed
white noise sequence according
𝑌[𝑛] = 𝑁[𝑛] + 𝑏1 𝑁[𝑛 − 1]
Where b1 is a real constant. Find the autocorrelation of Y[n] and the power density spectrum of
white noise and Y[n].
12
An Amplifier has three stages for which Te1=150 K (First stage),
[10]
Te2 = 350K and Te3 = 600K (Output stage). Available power gain of the first stage is 10 and overall
input effective noise temperature is 190K.
a) What is the available power gain of the second stage?
b) What is the cascade’s standard spot noise figure?
c) What is the cascade’s operating spot noise figure when used with a source of noise
temperature Ts= 50K?
[10]
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Total marks
[100]

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