MAHAKAVI BHARATHIYAR COLLEGE OF ENGINEERING AND TECHNOLOGY PREPARED BY R.SIVAKUMAR AP/MATHS QUESTION BANKDEPARTMENT: ECESEMESTER: IVSUBJECT CODE / Name: MA 2261/PROBABILITY AND RANDOM PROCESS UNIT–I: RANDOM VARIABLESPART -A (2 Marks)1. The CDF of a continuous random variable is given by0 ,x0, , Find the pdf and mean ofXF (x )x1 e,0 x(AUC Nov/Dec 2011)(AUC Apr/May 2011)2. The probab ility that a man shooting a target is 1 / 4. How many times must he fire so that the probability of his hitting the target atleast once is more than 2 / 3?(AUC May/Jun 2012)2n3.Find C, if PXnc;n 1, 2,.... (AUC May/Jun 2012) 3 4.A continuous random variable X has probability d ensity function3x2 , 0 x1 .Find K such that PXk0.5 f (x)0, otherwise(AUC Nov/Dec 2010)5.If X is uniformly distributed in (2 , 2 ) . Find the probability distribution function of=(AUC Nov/Dec 2010)6.Establish the memory less property of the exponential distribution.(AUC Apr/May 2011)7.If the probability density function of a random variable X iff (x )xin 0x 2,findP (X 1.5 /X 1). (AUC Apr/May 2010)2MA 2261 Probability and Random process IV Sem ECE –R.SIVAKUMAR Asst.Prof./MATHS 1
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6. Find the MGF of the random variable X having the probability density function
3 f ( x)
e
40
-x 2 , x 0
Also deduce the first four moments about the origin.
, elsewhere
(AUC Nov/Dec 2010)
7. If X is the uniformly distributed in (-1,1),then find the probability density function of
Y sin
x
(AUC Nov/Dec 2010) 2
8. If X and Y are independent random variables following
N (8, 2) and N (12, 4 3) respectively,findthe valueof such that (AUC Nov/Dec 2010)
P 2 X Y 2 P X 2Y
9. The probability mass function of random variable X is defined as
P ( X 0) 3c2 , P ( X 1) 4c 10c
2 , P ( X 2) 5c 1, Where c 0, and
P ( X r ) 0 if r 0,1, 2.Find ( AUCApr / May2010)
(i ) The value of c.
(ii ) P (0 X 2 / X 0)
(iii ) Thelargest value of X for which F ( X ) 1
2
10. If the probability that an applicant for a driver’s license will pass the road test onany given trial is 0.8. What is the probability that he will finally pass the test
(i) On the fourth trial and
(ii) In less than 4 trials? (AUC Apr/May2010)
11. The marks obtained by a number of students in a certain subject are assumed to
be normally distributed with mean 65 and standard deviation 5. If 3 students are
selected at random from this group, what is the probability that atleast one of them
would have scored above 75? (AUC Apr/May2010)
(AUC Apr/May 2011)
12. The Probability distribution function of a random variable X is given by
x , 0 x 1
f X ( x ) k (2 x ) ,1 x 2
0 , otherwise (AUC Apr/May 2011)
(i)Find the value of k , (ii)Find P (0.2 x 1.2)
(iii) What is P (0.5 x 1.5 / X 1)
(iv)Find the distribution functionof f ( x).
MA 2261 Probability and Random process IV Sem ECE – R.SIVAKUMAR Asst.Prof./MATHS3
2 x y , 0 x 1, 0 y 1 f ( x , y) . Find the correlation coefficient between X
0 , otherwise
and Y. (AUC Nov/Dec 2011)
3. If X 1 , X 2 , X 3 ,... X n are uniform variates with mean=2.5 and variance=3
4 , use Central
limit theorem to estimate P (108 S n 12.6) Where S n X 1 X 2 X 3 ... X n.n 48.
(AUC Nov/Dec 2011)
4. If X and Y are independent with a common pdf(exponential) e
x
, x 0 and f ( x)
0 , x 0
e y , y 0
. Find the PDF for X-Y. (AUC Nov/Dec 2011) f ( y) 0 , y 0
5. Two random variables X and Y have the joint pdf given by
k (1 x2 y ) , 0 x 1, 0 y 1
XY ( x , y) 0 , otherwise
. Find
(i) The value of k,
(ii) Obtain the marginal probability density functions of X and Y
(iii) Also find the correlation coefficient between X and Y.
(AUC Nov/Dec 2010)
6. If X and Y are independent continuous random variables. Show that the pdf of
= + is given by ℎ( ) = f X (u ) f Y (u v ) dv . (AUC Nov/Dec 2010)
7. If Vi , i=1,2,3,…20 are independent noise voltages received in an adder and V is the
sum of the voltages received, find the probability that the total incoming voltage V
exceeds 105, using the central limit theorem. Assume that each of the random
variables Vi is uniformly distributed over (0, 10). (AUC Nov/Dec 2010)
8. If X and Y are independent Poisson random variables with respective parameters 1 and 2 . Calculate the conditional distribution of X, given that + = .
(AUC Apr/May 2011)
9. The regression equation of X and Y is 3 y 5 x 108 0. If the mean value of Y is 44
and the variance of X is 169
th of the variance of Y. Find the mean value of X and the
correlation coefficient. (AUC Apr/May 2011)
10. If X and Y are independent random variables with density function,
1, 1 x 2 y , 2 y 4 X , find the density function of ( x )
0, otherwise , f Y ( y) 6
0 , otherwise
Z=XY. (AUC Apr/May 2011)
MA 2261 Probability and Random process IV Sem ECE R.SIVAKUMAR Asst.Prof./MATHS7
1. What is random process said to be mean ergodic? (AUC Nov/Dec 2011)
2. If X (t ) is a normal process with (t ) 10 and C (t 1 , t 2 ) 16e t1t2 find the variance
of 10 − 6 . (AUC Nov/Dec 2011)
(AUC May/Jun 2012)
3. State the Postulates of a Poisson process. (AUC Nov/Dec 2010)
(AUC Apr/May 2011)
4. Consider the random process ( ) =( + Ф) where Ф is a random variable with
density function f ( ) 1 ,
. Check whether or not the process is wide
2 2sense stationary. (AUC Nov/Dec 2010)
5. Prove the first order stationary process. (AUC Apr/May 2011)
6. Define a wide sense stationary process. (AUC Apr/May 2010)
7. Define a Markov chain and give an example. (AUC Apr/May 2010) 9
8. The autocorrelation function of a stationary random process is R( ) 16 . 1 6 2
Find the mean and variance of the process. (AUC May /Jun 2012)
PART –B (16 Marks)
1. Show that the random process X (t ) A cos( t ) is wide- sense stationary, if A and are constants and θ is a uniformly distributed in (0,2π).(AUC Nov/Dec 2011)
2. The process X (t ) whose probability distribution under certain condition is given by
( at )n1
, n 1, 2,... 1
P (t ) n (1 at ) Find the mean and variance of the process. Is at , n 0
1 at the process first-order stationary? (AUC Nov/Dec 2011)
(AUC Nov/Dec 2011)
(AUC Nov/Dec 2012)
3. State the Postulates of a Poisson process and derive the probability distribution. Also
prove that the sum of two independent Poisson processes is a Poisson process.
(AUC Nov/Dec 2011)
4. If the WSS process X (t ) is given by X (t ) 10cos(100t ), where is uniformly
distributed over ( , ), Prove that X (t ) is correlation ergodic.
(AUC Nov/Dec 2010)
(AUC May/Jun 2012)
(AUC Nov/Dec 2012)
MA 2261 Probability and Random process IV Sem ECE – R.SIVAKUMAR Asst.Prof./MATHS9
5. If the process X (t ) : t 0is a Poisson process with parameter λ, obtain
P X (t ) n. Is the process first order stationary? (AUC Nov/Dec 2010)
(AUC Nov/Dec 2012)
6. Prove that a random telegraph signal process Y (t ) X (t ) is a wide sense stationary process
when ∝ is a random variable which is independent of ( ), assume that values -1 and +1 with
equal probability and R XX (t 1 , t 2 ) e2
t1 -t2 .
(AUC Nov/Dec 2012) (AUC Nov/Dec 2010)
7. A random process ( ) defined by ( ) = + ,-∞< t <∞ , where A and B are independent random variables each of which takes a value -2 with probability 1/3 and a value 1 with probability 2/3. Show that ( ) is
wide-sense stationary.
(AUC Apr/May 2011)
8. A random process has sample functions of the form ( ) = ( + ) where ω is constant. A is a random variable with mean zero and variance one and θ is a
random variable that is uniformly distributed between 0 and 2π. Assume that the random variable A and θ are independent. Is ( ) is a mean ergodic
process?
(AUC Apr/May 2011)
9. If X (t ) is Gaussian process with μ(t)=10 and C( t 1 , t 2 )=16 e t1t2 , find the probability that
(i) X(10)≤8
(ii) X (10) X (16) 4 . (AUC Apr/May 2011)
10. Prove that the interval between two successive occurrences of a Poisson processwith parameter λ has an exponential distribution with mean 1/λ.
(AUC Apr/May 2011)
11. Examine whether the random process ( ) = ( + ) is a wide sense
stationary random variable in ( , ). (AUC Apr/May 2010)
12. Assume that the number of messages input to a communication channel in an
interval of duration t seconds, is a Poisson process with mean λ=0.3. Compute,(1) The probability that exactly 3 messages will arrive during 10 seconds
interval.
(2) The probability that the number of message arrivals in an interval of
duration 5 seconds is between 3 and 7. (AUC Apr/May 2010)
13. The random binary transmission process
(t ) is a wide sense process with 0
mean and auto correlation function R( ) 1 T , where T is a constant. Find the
mean and variance of the time average of X (t ) mean-ergodic?
(AUC Apr/May 2010)
14. The transition probability matrix of a Markov chain X n , n=1,2,3,… having 3 states
0.1 0.5 0.4
1,2 &3 is P 0.6 0.2 0.2 and the initial distribution is
0.3 0.4 0.3
P (0.7, 0.2, 0.1) find (i ) P 2
3 and (ii ) P X
3 2, X
2 3, X
1 3, X
0 2
.
(AUC Apr/May 2010)
MA 2261 Probability and Random process IV Sem ECE – R.SIVAKUMAR Asst.Prof./MATHS10
1. State any two properties of linear time-invariant system. (AUC Nov/Dec 2011)
2. If
X (t ) and Y (t ) in the system Y (t ) h (u ) X (t u ) du are WSS process, how
are their auto correlation functions related. (AUC Nov/Dec 2011)
3. If Y ( t ) is the output of an linear time invariant system with impulse response ℎ( ), then find the cross correlation of the input function ( ) and output function ( ).
(AUC Nov/Dec 2010)
4. Define Band-Limited white noise. (AUC Nov/Dec 2011)
(AUC Apr/May 2011)
5. Find the system transfer function if a linear time Invariant system has an impulse
power spectrum S yy ( ) of the output if the system transfer function is given by
H ( )
1
2i . (AUC Nov/Dec 2010)
6. If Y (t ) A cos( 0t ) N (t ), where A is a constant, θ is a random variable with a uniform distribution in (-π, π) and N (t
)is a band-limited Gaussian White noise with
N 0 , for B 0
power spectral density S NN ( ) 2 n t e power spectra
0 , elsewhere
density Y (t ) . Assume that N (t ) and θ are independent.
(AUC Nov/Dec 2010)
(AUC Apr/May 2010)
7. A system has an impulse response h (t ) e t
U (t ) . Find the power spectral density
of the output () , corresponding to the input (). (AUC Nov/Dec 2010)
1
8. Consider a system with transfer function 1 j . An input signal with auto correlation
function m ( ) m2 is fed as input to the system. Find the mean and mean square
value of the output. (AUC Apr/May 2011)
(AUC May/Jun 2012)
9. A stationary random process X(t) having the autocorrelation function R XX ( ) A ( ) is
applied to a linear system at time t=0 where f ( ) represent the impulse function.
The linear system has the impulse response of h (t ) e bt
u (t ) where u (t ) represents the unit step function. Find RYY ( ). Also
find the mean and variance of ( ).
(AUC Apr/May 2011)
(AUC May/Jun 2012)
-t
10. A linear system is described by the impulse response h (t ) RC 1
e RC u (t ). Assume an
input process whose Auto correlation function is B ( ) . Find the mean and auto
correlation function of the output process. (AUC Apr/May 2011) 11. If N (t )is a band limited white noise centered at a c arrier frequency ω0 such that
N 0 , for B 0
. Find the auto correlation of N (t ). S NN ( ) 2 0 , otherwise
(AUC Apr/May 2011)
(AUC May/Jun 2012)
MA 2261 Probability and Random process IV Sem ECE – R.SIVAKUMAR Asst.Prof./MATHS16