Solving an underdetermined system
A lot of firms will ask questions that are variations on the following: Suppose you have a $p$ -dimensional parameter $x$ you want to estimate, and you have...
Quant Question Bank
Rotating regression data
We have $X \sim U[-2,2], Y \sim U[-1,1]$. If we apply rotation matrix
OLS on a desert island
Without any package imports, how would you compute OLS?
Bounding $R^2$
We have $X, Y, Z$, where $Z \sim X$ has $R^2=.1$ and $Z \sim Y$ has $R^2=.2$. Bound the $R^2$ of $Z \sim X + Y$.
Find next in binary search tree
For a binary search tree (BST), if you are at node $A$, how would you find the lexicographically next node after $A$? That is, how would you find the smalle...
Finding top eigenvector
How do you find the top eigenvector of a square matrix?
Rolling 6 dice
What is the probability the sum of 6 dice rolls is a multiple of 6?
How to regularize a neural network?
Solution This is just a basic question to query if you have basic familiarity with neural networks. You should mention techniques such as dropout,...
Streaming mean and variance
How to update mean and variance in a streaming way? By streaming we mean without retaining the individual samples, i.e. computing $\bar x_n$ from only $\bar...
Hyperplanes
You have points in 3D. How to find hyperplane that minimizes z-axis error? How to find hyperplane that minimizes orthogonal error? When would you prefer one ...
Checking if a number is $2^n$
How would you programmatically check if an integer is a power of $2$?
OLS when $X$ is too large.
What is OLS solution? How is it derived? How to compute with $n \gg p$, such that $X$ cannot be loaded in memory. Consider both when $p$ very small, but also...
Prob Gaussians exceed eachother
Probability $X > 3Y$ when they are iid Gaussians.
Drawing primes
If I draw 4 numbers without replacement from the first 16 primes, what is the probability the sum is even?
2 player game
Suppose two players play a game alternating turns. Player 1 goes first and has 50% chance of winning in that round. Player 2 goes second (if P1 doesn’t win) ...
Uniform on unit sphere
If we draw uniformly form the unit sphere, and call its coordinates $(X,Y,Z)$, what is the variance of $X$?
Product of uniforms
If $U_1, U_2 \overset{iid}{\sim} U[0,1]$, compute $Pr(U_1U_2 \le .5)$?
Two 6-sided die
Let $X,Y$ be the outcomes of rolling two iid six-sided dies. Compute $\mathrm{Pr}(X+Y < XY)$.
Sine integral
Compute $\int_\pi^{2\pi} \sin x dx$.
Ordering uniforms
Suppose you are going to be shown three iid uniform rv, $U_i \sim U[0,1]$. Your task is to after seeing $U_1$, assign it lowest, middle, or highest. Then see...
Gambler’s ruin
If $X_0=0$, and goes up or down one at each time step with equal probability, what is the probability $X$ hits $4$ before $X$ hits $-6$?
OLS coefficient bounds
If $Y \sim X$ gives coefficient $\hat\beta$, what is the bound on the coefficient $\hat\theta$ from $X \sim Y$?
Probability of pizza
I like to eat pizza on weekends. $\mathrm{Pr}(\text{eat pizza Sat}) = .3$ and $\mathrm{Pr}(\text{eat pizza Sun}) = .4$. What is the probability that I will e...
Partitioning square roots
How would you partition the square roots of the first 50 natural numbers $\sqrt 1, \sqrt 2, \dots, \sqrt{50}$ into two groups $A, B$, such that
Gold and silver coins
Suppose we play a game where we flip a gold coin 500 times and a silver coin 500 times. We get 3 dollars per gold head, and 1 dollar per silver head, getting...
First color out
We have a sack with 60 balls: 30 yellow, 20 red, and 10 white. If we draw randomly from the sack w/o replacement until all are drawn, what is the probability...
Counting passwords
If I have an 8 digit passcode, taking values $0-9$, what is the probability that a randomly generated passcode has exactly $5$ occurrences of the same digit?...
Seating bar patrons
Suppose you run a bar that has 25 seats at the bar. Patrons will enter and seat themselves according to the following rules. They will not sit next to someon...