ISI Delhi Interview

Can you provide me some Interview questions for preparation of ISI or Indian Statistical Institute B. Stat interview?

[pawan]

ISI or Indian Statistical Institute Interview is for 10-15 minutes with a panel of 3-4 professors. One has to solve one question out of 2 questions (one from Maths or stats and one from microeconomics).
Few ISI Interview Questions

Coin tosses

Movement in a grid

Statement of LMVT

When is a differentiable function increasing , prove it etc.

Definition of limit of a function

Limit of composition of functions , a question from sequences etc.

Drawing a graph , mentioning roots , extreme points etc.

A question about geometry, a ball rolling around the perimeter of a triangle , angle covered needs to bee found etc.

Whether derivative of a function is necessarily continuous? Counter example etc.
Prove that any prime number when divided my 30 gives remainder 1 or a prime number.

Explain: (p+q)C2 - pC2 - qC2

An equation: x^4 - 4x^3 + ax^2 + bx + 1 = 0. give its value without actually computing anything.


If f and g are continuous functions and f is differentiable everywhere while g is not, what can be said about f(x)*g(x) to which i promptly replied that it would be differentiable at those points at which g is not differentiable when f at those points becomes zero.Rest everywhere its diff. in the domains common to both f and g. Prove it.

Define Differentiability?

Prove How L.H.D =R.H.D.

A ‘success’ is defined as picking an ‘A’ such that the polynomial x 2 − Ax + 1 has at least one real root. Mr. X is picking A from uniform distribution over [0, 5]. Mr. Y is picking A ∈ [0, 5] with the probability distribution function fA(a) given by : fA(a) = 2a 25∀a ∈ [0, 5]. Which person has a higher probability of success?

The table below shows the consumption pattern of a consumer over a period of two years. It is known that the consumer spends all his income in both years.
Year-1 Year-2
Goods Price Goods Price
Good 1 100 100 120 100
Good 2 100 100 x 80

(a) If Year-2 bundle is preferred over Year-1 bundle, what can you say about consumption of second good in Year-2?
(b) If Year-1 bundle is preferred over Year-2 then what can you say about consumption of second good in Year-2?
(c) When will the Weak Axiom of Revealed Preference be violated?
(d) It is given that consumer prefers Year-1 bundle. Is there any inferior good in this case?

Suppose a monopolist faces constant returns to scale. Answer the following parts:
(a) Why he never produces on the inelastic side of the demand curve?
(b) Why does total revenue increase on the inelastic side as price increases?

There are two cities C1 & C2. Every citizen in both cities have concave increasing utility function.
(a) Initially a social planner is supposed to allocate an endowment among all the citizens. How does he/she allocate it? Explain.
(b) Suppose entire endowment is C1 but with social planner. Again the task is to distribute endowment in C1 and C2. But, now the transfer of 1 unit of endowment to C2 results in loss of v fraction. How does he/she allocate? Explain.
(c) In the above case, which of the region gets more part of endowment. Explain with the proof.
(d) What happens if the utilities are convex?

(a) Define concavity.
(b) Given a strictly concave utility function v(x, y) = U(x) + y. Find the optimal consumption bundle.

A professor has 20 hours to allocate between two PhD students. Let x1 and x2 be the time allocated to the two students. The utility of each student is as follows:
U1 = −|x1 − 2| and U2 = −|x2 − 8|
(a) Find the set of Pareto efficient allocations.
(b) What will be the optimal time allocation if the professor maximizes a social function given by : min{U1, U2}