Hall Ticket No Question Paper Code: AEC003
INSTITUTE OF AERONAUTICAL ENGINEERING
(Autonomous)
Four Year B.Tech III Semester End Examinations (Supplementary) January, 2019
Regulation: IARE R16
PROBABILITY THEORY AND STOCHASTIC PROCESSES
Time: 3 Hours Max Marks: 70
Answer ONE Question from each Unit
All Questions Carry Equal Marks
All parts of the question must be answered in one place only
UNIT I
1. Define following types of events. Simple events ii) Conditional events iii) Independent events
iv)Joint events with examples
Five men in a company of 20 are graduates. Three men are picked out of 20 at random.
What is the probability that all are graduates?
What is the probability of at least 1 is a graduate?
2. A and B are independent events then prove that
Ac and B are also independent.
Ac and Bc are also independent.
The chances of B and C becoming the G.M. of a company are in the ratio 4:2:3. The
probabilities that the bonus scheme will be introduced in the company if B and C become
G.M. are 0.3, 0.7 and 0.8 respectively. If the bonus scheme has been introduced, what is the
probability that A has been appointed as G.M.?
UNIT II
3. State and prove properties of distribution function. Discuss the method of defining a conditioning
event.
A random variable X has the following probability function shown in Table
Table 1
x 0 1 2 3 4
1/25 3/25 1/5 7/25 9/25
Find The distribution function of X. P and P X
4. Define exponential distribution and calculate mean of exponential random variable.
If X and Y are independent Poisson random variable show that the conditional distribution of X
given X+Y is a binomial distribution.
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UNIT III
5. Explain the joint moments of random variables.
Statistically independent random variables X and Y have moments m10=2, m20=14 and m11=-6.
Find second central moment 22.
6. Show that the variance of a weighted sum of uncorrelated random variables equals the weighted
sum of the variances of the random variables.
The life time of a certain brand of an electric bulb may be considered as a random variable with
mean 1200 and standard deviation 250. Find the probability using Central Limit Theorem that
the average lifetime of 60 bulbs exceeds 1250 hours.
UNIT IV
7. When does the time average converge to the ensemble average? Justify the answer. Briefly
explain about Gaussian random process.
A random process is defined as cos(wct+ where is a uniform random variable
.Verify the process is ergodic in the mean sense and auto correlation sense
8. Define strict sense stationary random process, auto correlation and cross correlation function of
a random process.
Consider two random processes B sin!t and Bcos!t-A sin!t where A and
B are uncorrelated, zero mean random variables with same variance and is a constant. Show
that and are jointly stationary?
UNIT V
9. State and prove Winner-Khinchine theorem.
The auto correlation of a stationary random process is given by RXX a 0. Find
the spectral density function.
10. Explain power spectrums for discrete-time random processes and sequences and state any two
properties of cross-power density spectrum.
A WSS random process with autocorrelation function where a is a real positive
constant, is applied to the input of an LTI system with impulse response Where
b is a real positive constant. Find the autocorrelation function of the output of the system.