Rolling 6 dice
What is the probability the sum of 6 dice rolls is a multiple of 6?
Posts by Tag
- probability 10
- linear regression 7
- mental math 7
- optimization 3
- computational linear algebra 3
- linear algebra 3
- cross validation 2
- combinatorics 2
- maximum likelihood 2
- calculus 2
- lasso 1
- bootstrap 1
- autoregression 1
- method of moments 1
- programming 1
- bit manipulation 1
- principle components analysis 1
- algorithms 1
- machine learning 1
- data structures 1
- geometry 1
- PCA 1
- extracts 1
probability
Prob Gaussians exceed eachother
Probability $X > 3Y$ when they are iid Gaussians.
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 $\mathrm{Pr}(X+Y < XY)$.
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...
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...
linear regression
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 $Z \sim X + Y$.
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 ...
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...
OLS coefficient bounds
If $Y \sim X$ gives coefficient $\hat\beta$, what is the bound on the coefficient $\hat\theta$ from $X \sim Y$?
Estimating Variances in Autoregression
I was asked this in a quant research onsite interview. It is a question about estimating variances in a noisy autoregression setting.
mental math
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 $X$?
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 $\int_\pi^{2\pi} \sin x dx$.
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 $-6$?
optimization
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...
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,...
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
computational linear algebra
OLS on a desert island
Without any package imports, how would you compute OLS?
Finding top eigenvector
How do you find the top eigenvector of a square matrix?
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...
linear algebra
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...
Finding colinear subset
This note explores the following quant interview question: if you are given a matrix $X \in \mathbb{R}^{n\times p}$ that you are told has colinearity, how wo...
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 ...
cross validation
CV tradeoff
In a quant interview last year, I was asked a question about the tradeoff in changing the number of folds $k$ in $k$-fold cross-validation. Obviously, as the...
Estimating the SE of Lasso CV coefficients
This note comes from the following question: You have data $X, Y$ and you run a lasso-penalized linear regression of $Y$ onto $X$, with the lasso parameter t...
combinatorics
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?...
maximum likelihood
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...
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...
calculus
Product of uniforms
If $U_1, U_2 \overset{iid}{\sim} U[0,1]$, compute $Pr(U_1U_2 \le .5)$?
Sine integral
Compute $\int_\pi^{2\pi} \sin x dx$.
lasso
Estimating the SE of Lasso CV coefficients
This note comes from the following question: You have data $X, Y$ and you run a lasso-penalized linear regression of $Y$ onto $X$, with the lasso parameter t...
bootstrap
Estimating the SE of Lasso CV coefficients
This note comes from the following question: You have data $X, Y$ and you run a lasso-penalized linear regression of $Y$ onto $X$, with the lasso parameter t...
autoregression
Estimating Variances in Autoregression
I was asked this in a quant research onsite interview. It is a question about estimating variances in a noisy autoregression setting.
method of moments
Estimating Variances in Autoregression
I was asked this in a quant research onsite interview. It is a question about estimating variances in a noisy autoregression setting.
programming
Checking if a number is $2^n$
How would you programmatically check if an integer is a power of $2$?
bit manipulation
Checking if a number is $2^n$
How would you programmatically check if an integer is a power of $2$?
principle components analysis
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 ...
algorithms
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...
machine learning
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,...
data structures
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...
geometry
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 $Z \sim X + Y$.
PCA
Finding colinear subset
This note explores the following quant interview question: if you are given a matrix $X \in \mathbb{R}^{n\times p}$ that you are told has colinearity, how wo...
extracts
Store-bought extracts for at-home drinks
I have recently been watching videos from Darcy O’Neil’s YouTube channel Art of Drink. In it, he goes through techniques for how to extract flavor compounds ...