Birla Institute of Technology and Science – Pilani, Hyderabad Campus
Second Semester, 2020-2021
COMPREHENSIVE EXAMINATION
(Open Book)
Course Title: Probability & Statistics
Time Duration: 120 minutes
Max. Marks: 40
Course No.: MATH F113
Date: 13.5.21
Day: Thursday
Instructions: Answer all questions with properly defined statements; Marks for each question are mentioned at the end
of the question; Symbols carry the usual meaning as in the text book. WRITE YOUR NAME AND ID SUITABLY ON
THE TOP SHEET AND EACH SUBSEQUENT PAGE. SCAN YOUR ANSWER SHEET AND UPLOAD IT IN
GOOGLE CLASSROOM AS A SINGLE PDF FILE. FILE NAME SHOULD BE YOUR ID.PDF. USE THE
APPROPRIATE DISTRIBUTION TABLES AS HAVE BEEN PROVIDED TO YOU. YOU MAY USE
CALCULATORS.
DO NOT LOG OUT OF METTL PLATFORM BEFORE SUBMITTING YOUR ANSWER SCRIPTS.
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1. (a) Let 𝐴, 𝐵 be events with probabilities 𝑃(𝐴) = 4 and 𝑃(𝐵) = 3 . Show that 12 ≤ 𝑃(𝐴 ∩ 𝐵) ≤ 3 . [4M]
(b) Suppose 𝐴, 𝐵 are two events, such that 𝑃(𝐴) = 0.5, 𝑃(𝐵) = 0.4 , 𝑃(𝐴 ∩ 𝐵) = 0.25. What is the
probability neither of the events happen?
[2M]
2. (a) Three balls are randomly chosen from an urn containing 3 white, 3 red and 5 black balls. Suppose
that we win Rupee 1 for each white ball selected and lose Rupee 1 for each red ball selected. For a black
ball selected, the prize money is 0. If we let 𝑋 denote our total winnings from the experiment, then
answer the following:
(i)
What are the possible values that 𝑋 can take?
(ii)
Find the probabilities associated with all the different values of 𝑋.
[2M+4M]
(b) It is known that screws produced by a certain company will be defective with probability 0.01
independently of each other. The company sells the screws in packages of 10 and offers a
money-back guarantee that at most 1 of the 10 screws is defective. What proportion of packages sold
must the company replace?
[4M]
(c) Suppose that 𝑋 is a continuous random variable whose probability density function is given by,
𝑓(𝑥) = 𝑐(4𝑥 − 2𝑥 2 ), 0 < 𝑥 < 2
= 0 , otherwise
Find the value of 𝑐 and the probability 𝑃(𝑋 > 1).
3.
[4M]
Let 𝑋 be a normal random variable with variance 9. Using a sample of size 10 with mean 𝑥̅ , test the
hypothesis 𝜇 = 𝜇0 = 24 , against the three kinds of alternatives,
(a) 𝜇 > 𝜇0
(b) 𝜇 < 𝜇0
(c) 𝜇 ≠ 𝜇0 .
[In each case, clearly mention the rejection region in terms of 𝑥̅ ]
[Take the level of significance to be 𝛼 = 0.05]
[6M]
4.
Researchers are interested in the mean age of a certain population. A random sample of 10 individuals
drawn from the population of interest has a mean of 27. Assuming that the population is normally
distributed with variance 20, test the hypothesis 𝐻0 : 𝜇 = 30 against 𝐻𝑎 : 𝜇 ≠ 30. Take the level of
significance 𝛼 = 0.05.
[2M]
5. Consider the data,
𝑥
112.3
97
92.7
86
102
99.2
95.8
103.5
89
86.7
𝑦
75
71
57.7
48.7
74.3
73.3
68
59.3
57.8
48.5
Obtain the equation of the least squares line.
[8M]
6. Find the mean and variance of the uniform distribution, the pdf being given by,
1
𝑓(𝑥) = 𝑏−𝑎 , 𝑥 𝜖 [𝑎, 𝑏]
= 0 , otherwise
[4M]
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