ISI MStat Interview 2024 Questions

The questions posted here were contributed by various members of the MStat 2024-2026 batch, and are not necessarily interview questions, but certainly interview-oriented. Special thanks to Rinika Jana (LinkedIn) for compiling everyone's inputs.

It is very important to remember that the interview questions are intended to probe the depth of one's understanding, and are often asked based on one's topic preferences. It is not compulsory nor expected to be able to flawlessly answer every single question below. But being proficient in at least 2-3 topics will help.

Inference

1. Given $X \sim \Bin(n,p)$, $Y \sim \Bin(m,p)$. The ultimate goal is to find what $X | X+Y$ follows. Here are some milestones --
• What does $X+Y$ follow?  Ans: $\Bin(m+n, p)$.
• Is this always true? Do we need something more?  Ans: Independence of $X$ and $Y$.
• Once the pmf is obtained for the distribution of $X|X+Y$, why is the pmf obtained independent of parameter $p$?
• Give a real life example of the need of testing
  $H_0: p=p'$ vs $H_1: p>p'$.
• Using samples $X_1,\dots,X_n \sim \Ber(p)$ and $Y_1,\dots,Y_m \sim \Ber(p')$ and the expression of the pmf you have derived, suggest a test statistic to conduct the above test.
• Why are you choosing this statistic?
• What is the critical region and why?
2. Given $X_1, \dots, X_n \sim iid \, \N(\mu, 1)$.
• What is the value of $\gamma = \Pr (X_1 > 1)$?
• What is the MLE of $\gamma$ based on all $n$ observations?
• Based on 1 observation?
• What is the u.e. (unbiased estimator) of $\gamma$ based on all $n$ observations?
• Based on 1 observation?
• Intuitively, which is the better estimator between the MLE and the UE?
• The MLE of $\gamma$ seems to contain the $\Phi^{-1}$ function. So prove that $\Phi$ is invertible.  Note: Real Analytic question.
3. Given $X$ and $Y$ random variables with $\E (X) = \E (Y) = 0$. Define:
   $W=X+Y$ and $V=X-Y$, given that $W \perp V$.
• What can you say about $\Cor (X,Y)$?
• Construct a random vector $(

Linear Models

1. 
   

Combinatorics

1. Let $A = \{1, 2, \dots, 50 \}$ and $B = \{ 1, 2, \dots, 100 \}$.
• How many functions are possible from $A$ to $B$?
• How many of them are onto?
• How many of them are one-one?
• A point $x$ is called a "fixed point" of $f$ if $f(x) = x$. Suppose we decide to plot a random function $f$ from amongst all one-one functions. Define
  $X: \text{ number of fixed points of } f$.
• What is the support of $X$.
• Find the probability mass at $X=100$. What about $X=99$?  Note: these will be hard, and go to show that the exact distribution of $X$ is non-trivial.
• Find $\mathbb E (X)$.  Hint: Try to break $X$ as a sum of 50 indicator functions; define them properly.

Linear Algebra

1. What is an eigenvalue? What is an eigenvector? What is a characteristic polynomial?
• Do non-square matrices have an eigenvalue?
2. Let $A$ be a square matrix of order 3.
• Using characteristic polynomial, find the inverse of $A$.
• Using what you have derived, justify the case when $A^{-1}$ does not exist.
• Can $A_{3 \times 3}$ have complex roots?
• Suppose $A^{-1}$ exists and $A = A^4$. Then find an eigenvalue of $A$.
3. Given a system of equations $A \vec{u} = 0$, with orthogonal $A$ and $\vec u = (x, y, 1)'$.
• What can you say about the solution of $\vec u$?
• Suppose $A$ is now just non-singular instead. What can you say about $\vec u$?
• Suppose $A$ is now a singular matrix. Can you say anything about the solution of $\vec u$?

Real Analysis

1. Give an example of a function $f: \RR \to \RR$ which is discontinuous at all points, but $|f|$ is continuous everywhere.

Contributors

Big thanks to all contributors for sharing their valuable experiences. Some names are:

Sukanya, Eshan, Raghav...