Random process and linear algebra

ST.ANNE’S
COLLEGE OF ENGINEERING AND
TECHNOLOGY
(An ISO 9001:2015 Certified Institution)
Anguchettypalayam, Panruti – 607106.




QUESTION BANK (R-2021)



MA3355
RANDOM PROCESSES AND LINEAR
ALGEBRA




V.PRAKASH,MS.c,M.Phil,B.Ed,
Associate Professor
Department Of Mathematics Page 1

, ST.ANNE’S
COLLEGE OF ENGINEERING AND TECHNOLOGY
(An ISO 9001:2015 Certified Institution)
Anguchettypalayam, Panruti – 607106.

QUESTION BANK
PERIOD: AUG -DEC-2022 BATCH: 2021 – 2025
BRANCH: ECE YEAR/SEM: II/03
SUB CODE/NAME: MA3355 –RANDOM PROCESSES AND LINEAR ALGEBRA


UNIT I – PROBABILITY AND RANDOM VARIABLES
PART – A
1. Define random variable.
2. X and Y are independent random variables with variances 2 and 3. Find the variance of 3X +4Y.
3. Let X be a R.V with E[X]=1 and E[X(X-1)]=4 . Find var X and Var(2-3X).
4. The number hardware failures of a computer system in a week of operations as the following pmf:
Number of failures: 0 1 2 3 4 5 6

Probability : 0.18 0.28 0.25 0.18 0.06 0.04 0.01

Find the mean of the number of failures in a week

5. A continuous random variable X has the probability density function given by f ( x)  3x 2 ,0  x  1 .
Find K such that P(X > K)= 0.5
Cxe x , x  0
6. A random variable X has the pdf f(x) given by f ( x)   . Find the value of C and c.d.f
 0, x  0
of X.
7. The cumulative distribution function of a random variable X is F(x)= 1  (1  x)e  x , x  0 . Find the
probability density function of X.
 0, x  2
1
8. Is the function defined as follows a density function? f ( x)   (3  2 x),2  x  4
18
 0, x  4
2 x,0  x  1
9. Let X be a R.V with p.d.f given by f ( x)   . Find the pdf of Y =(3X +1).
0, otherwise
 0, x  0
 x 2
10. Find the cdf of a RV is given by F ( x)   ,0  x  4 and find P(X>1/X<3).
 16
 1,4  x
11. A continuous random variable X that can assume any value between x = 2 and x = 5 has a density
function given by f(x) = K(1 + x). Find P[X<4].


V.PRAKASH,MS.c,M.Phil,B.Ed,
Associate Professor
Department Of Mathematics Page 2

, 12. The first four moments of a distribution about x = 4 are 1, 4, 10 and 45 respectively. Show that the
mean is 5, variance is 3,  3  0 and  4  26.
13. Define moment generating function.
 2
 3,x 1
 1
14. Find the moment generating function for the distribution where f ( x)   , x  2 .
 3
0, otherwise

15. For a binomial distribution mean is 6 and S.D is 2 . Find the first two terms of the distribution.
16. Find the moment generating function of binomial distribution.
17. The mean of a binomial distribution is 20 and standard deviation is 4. Find the parameters of the
distribution
18. If X is a Poisson variate such that P(X=2) = 9P(X=4) +90P(X=6),find the variance.
19. Write the MGF of geometric distribution.
20. One percent of jobs arriving at a computer system need to wait until weekends for scheduling ,
owing to core-size limitations. Find the probability that among a sample of 200 jobs there are no
job that have to wait until weekends.
1
21. Show that for the uniform distribution f ( x)  ,a  x  a the moment generating function about
2a
sinh at
origin is .
at
22. If X is a Gaussian random variable with mean zero and variance  2 , find the probability density
function of Y  X .
e  x , x  0 1
23. A random variable X has p.d.f f ( x)   . Find the density function of
 0, x  0 x
24. State Memoryless property of exponential distribution.

25. The mean and variance of binomial distribution are 5 and 4. Determine the distribution.
26. For a binomial distribution mean is 6 and S.D is √𝟐 . Find the first of the distribution.
27. What are the limitations of Poisson distribution.
28. A random variable X is uniformly distributed between 3 and 15.Find mean and variance.
𝟑 𝟐 ),
29. A continuous random variable X has a p.d.f given by 𝒇(𝒙) = {𝟒 (𝟐𝒙 − 𝒙 𝟎<𝑥<2
.Find 𝒑(𝒙 > 1)
𝟎 , 𝒐𝒕𝒉𝒆𝒓𝒘𝒊𝒔𝒆

30. Let X be the random variable which denotes the number of heads in three tosses of a fair coin.
Determine the probability mass function of X.
𝒌𝒆−𝒙 , 𝒙 > 0
31. If 𝒇(𝒙) = { is p.d.f of a random variable X ,then the value of K.
𝟎, 𝒐𝒕𝒉𝒆𝒓𝒊𝒔𝒆
𝟏
,𝒙 = 𝟎
32. Find the mean and variance of the discrete random variable X with the p.m.f 𝒑(𝒙) = {𝟑𝟐
,𝒙 = 𝟐
𝟑




V.PRAKASH,MS.c,M.Phil,B.Ed,
Associate Professor
Department Of Mathematics Page 3