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Name of the University : IEOR @ IIT Bombay Industrial Engineering & Operations Research Indian Institute Of Technology Bombay
Exam : Ph.D. Admissions Entrance Test
Document Type : Model Question Paper
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IEOR – IITB Ph.D. Admissions Entrance Test Question Paper

** Following are some model questions for admissions-tests.
** The pattern and contents may be updated ocassionally.

Related : Indian Institute of Technology Bombay PhD Entrance Examination Sociology & Pschology Question Paper :

** These question papers may be used for guidance but cannot be considered as a reference for the pattern, breadth and scope of the tests.


** No clarications on questions should be sought during the examination.
** Answer as many questions as you can.

** Multiple choice questions : Write the most appropriate choice (example: (Q)) clearly in the space provided on the right side. Each correct answer gets two points (2) and wrong choice gets one negative point (-1).

** some of which have appeared in the previous written tests Candidates are to answer as many questions as possible

Sample Questions

Section I

Question 1 :
X, Y, Z and W are jointly distributed Bernoulli random variables; and each of these can assume values 0 or 1 only. It is known that X = max {W, Z} and Y = min {W, Z}.
(a) If E(X) = 0.6 and E(Y) = 0.1, determine the Conditional Expectation E(X | Y = 0).
(b) Determine the maximum possible value of E(Y) if it is only known that E(X) = 0.6 (and there is no other numerical information available).

Question 2: Solve the following optimization problem using any algorithm:
Maximize x1 + x2
s.t. 3 x1 + 4 x2 = 11
x1 + 2 x2 = 4
x1 , x2 2 (set of positive integers)

Question 3 :
Sudoku is a popular puzzle that appears regularly in the daily newspapers. The puzzle is to fill in the girds in such a manner that every row, every column and every 3×3 box accommodates the digits 1 to 9, without repeating a digit. A sample puzzle is shown on the right side. Now, formulate an optimization model to solve the puzzle. Clearly define the variables, constraints and objective function. DO NOT SOLVE THE PUZZLE.

Question 4 :
Weekly demand for an item stocked by a retailer is uncertain. You are given past data (say demands d(t) in week t for t = 1, 2, …, T – current time, are known),
(1) Explain how you would decide on the stock level that a retailer would keep (assume that the item is ordered and replenished at the beginning every week). You would need to assume relevant parameters and you need to state the decision model clearly. If you were to do this to minimize costs, what would your approach be? If you were to do this to meet customer service requirements, what would your approach be? How would you reconcile different policies that arise from these two approaches?

(2) The previous setting is for a fixed price p of the item. Someone suggests that demand in week t depends on the price p(t) that you charge in week t. Give an example of a function that captures this dependence of demand on price, with appropriate assumptions. What are the ways in which the correct price p(t) can be set in this setting (i.e. suggest an optimization problem to determine the price p(t), as usual with some assumptions).

Question 5 :
Let X1, X2 … Xn, be IID random sample from a normal population having mean µ and variance s2. Let X and S2 denote the sample mean and sample variance, respectively. Determine E[S2]. Show all steps in your computation.

Question 6 :
For the Markov chain {Xn, n = 0} on {1, 2, 3, 4, 5} with P as its transition matrix, find lim n n ij p ?8 for all state-pairs (i, j) with X0 = 2

Section II

Part A :
Multiple choice questions: Write the most appropriate choice (example: (Q)) clearly in the space provided on the right side. Each correct answer gets two points (2) and wrong choice gets one negative point (-1).
1. Suppose a fair coin is tossed repeatedly and heads appeared in the rst 1000 tosses. What is the probability that tail will turn up in the 1001th toss?
(P) 21001
(Q) 0.5
(R) 0
(S) 1 Ans.

2. Consider a square of side 1 metre. A number of ants are dropped on the edges of the square at time t = 0. Once an ant is dropped, it starts walking along the edge (assume that the ants walk only along the edges) with a speed of 1 metre per minute. Once it reaches a corner, it falls o the square. If two ants collide head on, they immediately reverse directions and continue walking with the same speed. The earliest time after which one can be sure that all ants fall o the square is
(P) 1 min
(Q) 3 min
(R) p 2 min
(S) never Ans.

3. Laila and Majnu go to a dinner party with four other couples. Each person there shakes hands with everyone he or she doesn’t know. Later, Majnu does a survey and discovers that every one of the nine other attendees shook hands with a dierent number of people. How many people did Laila shake hands with?
(P) 8
(Q) 1
(R) 7
(S) 4 Ans.

4. Five people of dierent heights go to a restaurant and sit at a round table at random. The probability that they are sitting in the order of their heights is
(P) 1120
(Q) 1 12
(R) 1 2
(S) 1
10 Ans.

5. Let f : R ! R be a twice-dierentiable function. Suppose that x is a local minimum of f. Which of the following is true always?
(P) f00(x) > 0
(Q) f0(x) > 0
(R) f00(x) < 0
(S) None of the above

6. Suppose that there are doors numbered 1 to 100 in a hallway. One person comes and opens all doors. A second person comes and closes all even numbered doors. Then a third person comes and changes the state of all doors that are multiples of 3. This continues until 100 people have passed the hallway. Which of the following doors remain open eventually?
(P) Door no. 50
(Q) Door no. 36
(R) Door no. 13
(S) None of the above Ans.

7. Suppose a machine produces bolts, 10% of which are defective. What is the probability that a box of 3 bolts contains at most one defective bolt
(P) 1
(Q) 0.872
(R) 0.972
(S) 1 4 Ans.

8. Suppose a circle with centre at O and a line l intersects at a point P and nowhere else. Then the angle between OP and l is
(P) can not be determined from given data
(Q) 180
(R) 90
(S) 45 Ans.

9. When we write 23.723723723723…, we are writing a representation of
(P) an irrational number
(Q) a nonterminating, nonrepeating decimal
(R) a rational number
(S) a binary number Ans.

10. Following table (Table 1) shows the results of a hypothetical experiment in which a 6-sided die is tossed 6000 times. What, if anything, is wrong with this table?
(P) There is nothing wrong with the table; it is entirely plausible.
(Q) The data for the absolute frequency and the cumulative absoute frequency are in the wrong columns.
(R) There is no way a coincidence like this could ever occur
(S) The cumulative absolute frequency values do not add up right.

11. Let f : R ! R2 be a linear function which is not constant. What is the number of zeros of f?
(P) 1
(Q) 2
(R) 0
(S) innite Ans.

12. Suppose you are sampling data from a Normal distribution with unknown mean. You have good estimates of the mean and standard deviation of the sample data. Which of the following can be determined on
the basis of this information?
(P) The 50% condence interval of the true mean.
(Q) The 90% condence interval of the true mean.
(R) Both the above.
(S) None of the above. Ans.

13. Suppose the average weight of a given set of 100 persons is 60 kg, and their average height is 160 cm. Consider the three statements below:
(i) There must exist at least one person weighing at most 60 kg.
(ii) There must exist at least one person weighing at most 60 kg and having height at most 160 cm.
(iii) There must exist at least one person weighing at most 120 kg and having height at most 320 cm.

Which of the four alternatives below are true?
(P) Statement (i) is true and the other statements are false.
(Q) Statements (ii) and (iii) are true, and (i) is false.
(R) Statement (iii) is true, and the others are false.
(S) Statements (i) and (iii) are true, and (ii) is false.

15. Two fair coins A and B, made of dierent materials, are tossed simultaneously. Due to dierences in the material of the coins, coin A lands rst in 90% of the experiments. If coin A lands rst, the coin B always shows the same face as coin A. Else, the outcomes are independent. What is the probability that both coins show the same face?

Part B :
A1 Consider the function L(x) : R ! R dened as L(x) := maxfa1x + b1; : : : ; anx + bng for a given 2n reals fa1; : : : ; an; b1; : : : ; bng.
1. Is L() a continuous function?
2. Is L() a dierentiable function?
3. Give sucient conditions under which it is a dierentiable function. Are these conditions also necessary?

4. A real valued function f() on real line R is said to be convex if for any two given reals x and y, we have f(x + (1 ?? )y) f(x) + (1 ?? )f(y) for all 2 (0; 1). Is L() a convex function?
5. A function f() : R ! R is said to be a concave function if ??f() is a convex function. Is L() a concave function?

6. Give sucient conditions under which L() is a concave function.
7. Give sucient conditions under which L() is both a convex and a concave function.
8. A set A R2 is a convex set if for any given pair of points x; y 2 A, we have point x+(1??)y 2 A for all 2 (0; 1). Identify a convex set in R2, if any, when function L() is plotted.

Part C :
B1 Let A1;A2; : : : ;An be independent events dened on a sample space such that 0 < P(Aj) < 1 for j = 1; : : : ; n (here, P(Aj) denotes the probability that event Aj occurs). Then show that must contain at least 2n events.

B2 Let X be uniformly distributed over the interval [??1; 1]. Let Y be dened as Y = X2.
(a) Are X and Y independent random variables?
(b) Compute E[XY ], the expected value of XY . Also, compute E[X]E[Y ], and verify if E[XY ] = E[X]E[Y ].
(c) Comment on the above observations.

B3 Let random() be a function that takes an integer n 1 as input, and outputs a random integer from f1; : : : ; ng. Consider the following pseudo-code:
x = 0;
do while (N > 1)
N = random(N);
x = x + 1;
end do
Let E(n) denote the expected value of the result of the above pseudocode with initialization N = n.

1. Compute the values of E(1) and E(2).
2. Derive a recurrence relation that will help you compute E(n) when E(1); : : : ;E(n ?? 1) are given.
3. Compute E(n) ?? E(n ?? 1).
4. Using part 3, determine a closed-form formula for E(n).

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