RTU 2016 Question Paper Semester IV Electronics and Communication Engineering Random Variables & Stochastic Processes

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RollNo. TotalNo of Pages:fl
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CN 4F41.3!
t-.{ B.Tech. IV-Sem (Main & Back) Examl June-July 2016
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rll Electronics & Communication
+ 4EC2A Random variables & stochastic processes

Time: 3 Hours Maximum Marks: 80
Min. Passing Marks (Main & Back): 26
Min. Passing Marks (Otd Back): 24
I nstructions to Candidate s : -

Attempt any five questions, se:lecting one question
from each unit. All
Questions carry equal marks. schematic diagrams musi be shown
wherever necessary. Any data you feel missing suitably be assumed and
stated clearly.
units of quantities used/ calculated must be stated clearly.
use of following supporting material is permitted during examination.
(Mentioned in form No.205 )
1. NIL 2. NIL

Q.1 (a) Prove that 2" - (n +1) equations are needed to establish the mutual independence
of n - events.
(b) The age of a person when he dies is denoted by t. The probability that ,
given by the following equation = J:l

e (, <,s)='fo f,lo,

Where A(t) is a function determined from mortality records. The curve between
A(t) and t is given in Fig.l(b) for 0 < t < I00 years and A(t) is given as
A(t) = 3 x l0-e t' 1100 - t)2; 0 < t < l00years.
Determine the probability that a person wiil die between the ages of 60 & 70
assuming that he was alive at 60.
tgl

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OR
Q.1 (a) In a system, there are n components connected in series. This system works
successfully when all units (components) work successfully. The operation of
each component is independent to each other. The probability of successful
operation of the components is p1 where i = 1,2,3,. ..., n. Find the
probability that the system functions satisfactorily. t8l

i,n,,*ffiI?l-oo,nu,
fig.1(a)
(b) State & explain the theorem of total probability & Bayes Theorem. t8l

Q.2 (a) Explain all the properties of conditional Distribution. t6l
(b) Determine the mean and variance of the random variable X of the following
(i) Uniformdistribution tsl
(ii) Exponential distribution lsl

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OR
Q'2 (a) Determine the mean and variance of the random variable X of the following
(i) Normaldistribution t5l
(ii) Rayleigh distriburion lsl
(b) Prove the reproductive property of independent Poisson Random Variable.
Hence find the probability of 5 or more telephone calls arriving in a 9min. period
in a collage switch board, if the telephone calls that are arrived at the rate of 2
every 3min. Follow a Poisson distribution.
t6l
UNIT-III
Q.3 (a) Consider Z = X + Y, show that if X and Y are independent Poisson's RV's with
parameters ),1 and 1,2, respectively, thenZ is also a Poisson Random Variable.
[8]
(b) Let X and Y be the independent random variables with common parameters 1,.
Define U = X + Y, V = { - Y. Find the joint and marginal pdf of U and V.
t8l
OR
Q.3 (a) A voltage V is a function of time ilO i. given by
V(0=Xcoswt+Ysinwt
In which w is a constant angular frequency and X = y = N(0, o2) and they are
independent.
(i) Show that v(t) may be written as
V(t)=Rcos(wt-0)
(ii) Find the pdfs of RV's R and 0 and show that R and 0 are independent.
(b) Define a two dimensional random variable. Give an example of the out - come of
a random experiment, that is a two dimensional random variables.
t6l

Q.4 (a) Consider a continuous random variable X, prove that
Elrl = or(*)la^-
J-h l,ry(,.)ox
. Where Fx (x) is the cdf of X.
t6l
(b) Explain the followings;
(i) Liapounoff's form of CLT. tsl
(ii) Lindberg - Levy's form of CLT. tsl
Where CLT = Central Limit Theorem.

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OR

Q.4(a)Considertherandomvariablexwtrosecharacteristicsfunctionisgivenby

ox(w)=
,-,;l;l:l
ti t8l
Determine the Pdf of X'
(b)TheMomentgeneratingfunctionofarandomvariableXisgivenby
5
Mx (o) =r:E2
of X' t4l
Determine the standard deviation
t4l
(c)WritedownallthepropertiesofcharacteristicsfunctionQ*(w),
UNIT.V
tSl
prorni-"r-.f-Po;rr to:::l1l1l.1tj,t'^-
Q.s (a) write and explain au the -,.,.by
(b) Let X(t);.r-i; p.o".r, *ith the auto correlation function given
rjs
n*" (t) =l 19- l"o'(*o')

WhereAoandwoareconstants.DeterminethepsdofX(0.t8]
OR
Q.5(a)In.thefiguregivenbelow,X(t)beainputvoltagetoacircuitandY(t)bethe
outputvoltage.Theprocessx(t)isastationaryrandomprocesswithzeromean
and auto correlation
/\
Raa (t) = -0lTl
e
Determine E[Y(t)], Sw(w) and Rvv(t)'

Fis.5(a)

(b) The psd of white noise tY)is 6 x 10-6 *, Hz., is applied
to an ideal Low Pass

bandwidth w rad/sec' Find w so
that
Filter with power transfer function 1 and t6l
output average noise power is 15
watt'

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