No. of Pages: 3 A

APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY
FIRST SEMESTER M.TECH DEGREE EXAMINATION, APR 2021/DEC 2021

Branch: Mechanical Engineering
Streams: (i) Industrial Engineering (ii) Financial Engineering

0113146017 Probability and Stochastic Processes

Max. marks: 60 Duration: 3 hours

PART A
(Answer any two questions. Each question carries 9 marks.)

1. Suppose random variables X and Y have joint probability density given by:

2, 021321, x+y0 otherwise

S(x,y) = |

(a) Find the marginal densities
(0) Find P(X > 37 = +) and E[X|Y]

2. Let X and Y be independent random variables with
1 1
P(X ಎ0) = P(X = I= ‏نہ‎ PY = 0) = ۶)۴ = 2) = 5

(a) Compute the joint distribution X and Y.
(b) Compute the probability distribution of Z = X + Y

3. (a) X and ¥ are discrete random variables with joint probability mass function

X=1 0.25 0.25 0
2

X=2 0 0.25 0.25

Find the correlation coefficient of X and Y and interpret the result.

(b) Suppose that the service time (in minutes) for a customer at a bank counter is a random
variable with mean 2 and variance 1. Assume that service times for different customers
are independent. Using central limit theorem, find the probability that the total service
time for 50 customers is between 90 and 110 minutes.

PART-B

(Answer any two questions. Each question carries 9 marks.)

4. A rapid transit system has just started operating. In the first month of operation, it was found
that 25 % of commuters are using the system while 75 % are travelling by automobile.
Suppose that each month 10 % of transit users go back to using their cars, while 30 % of
automobile users switch to the transit system.

(4)
(5)

(3)
(6)

(5)

(4)