Citadel Securities Interview Questions

You regress X1 against Y and obtain an R-squared value of 0.1. Then, you regress X2 against Y and obtain an R-squared value of 0.2. What is the maximum possible R-squared when regressing both X1 and X2 against Y together?

Given a string, return all possible permutations of its characters as a list. You may use recursive backtracking or any relevant Python packages.

Starting at the origin on a grid, you can move either up/down or left/right at each step. After N steps, how far can you be from the origin? Additionally, if there are two boundaries, what is the probability that you reach one boundary before the other?

If you toss a fair coin N times, what is the probability that you will get an even number of heads? Follow-up: Now, what if you have N - 1 fair coins, and the Nth coin is biased and lands on heads with probability p?

What happens if two features are highly collinear in Lasso regression?

Explain the basics of logistic regression.

What happens to the R-squared (R^2) value if you double the data in a regression analysis? What about the t-value?

We have n switches, each with probability p_i of being on and 1-p_i of being off. What is the probability that C or more switches are on? What if C is a uniform random variable between 0 and n?

Given an array representing the price of a stock on each day, and an integer k, determine the maximum profit you can achieve by completing at most k buy-sell transactions (you must sell before buying again).

Given n balloons, each balloon is painted with a number on it represented by an array nums. You are to burst all the balloons. If you burst balloon i, you get nums[left] * nums[i] * nums[right] coins. Here, left and right are adjacent indices of i. After the burst, the left and right balloons become adjacent. Find the maximum coins you can collect by bursting the balloons wisely.

Randomly generate points (x, y) on the circumference of a circle in two dimensions. What is Var(x)? What if you randomly generate points on the surface of a sphere instead?

In ridge regression, what happens to the expected value of the estimate if you perturb your data matrix?

What is the limit of x^x as x approaches 0?

What is the difference between ordinary least squares (OLS) regression and Ridge regression?

Implement a Python function to price an American option using backward induction.

What is the difference between concurrent and multicore systems?

How does std::unordered_map work in C++? Provide an example.

Implement a thread pool for sending messages.

Reverse the words in a sentence string in-place.

Design a random number generator that, given a list of probabilities corresponding to a discrete set of outcomes, outputs each outcome with its specified probability.

Implement a shared pointer class in C++ that manages shared ownership of a dynamically allocated object. Ensure proper reference counting and memory management.

Implement a shared_ptr smart pointer in C++ that manages reference counting and ensures proper deletion of the managed object when no references remain.

Given 25 horses and a race track where only 5 horses can race at a time, what is the minimum number of races required to determine the fastest horse?

What is the square root of 800 rounded to two decimal places?

Write code to generate the Fibonacci sequence.

Given constraints on the correlations between three random variables, provide bounds for the correlation between one pair of them.

Describe a program to determine whether a given integer is a prime number.

What is the shape of the gamma curve for a call option?

What is the house edge of a standard game of roulette?

What is the probability that a coin is biased (always lands heads), given that you observe five heads in a row?