Citadel Interview Questions - Complete List

Quant Interview Questions

152 questions

What is covariance?

Quantitative Analyst Interview

Build a linked list in Java.

Software Engineer Interview

Implement a balanced binary search tree from scratch and explain its time complexity. Additionally, optimize the tree to efficiently handle duplicate values.

Software Engineer Interview

Given an array representing a sequence of user events (where each element is a user ID), find the length of the longest contiguous subarray where the most frequent user's frequency within the subarray equals the minimum frequency of any user in the entire event log. Implement an efficient function to return this maximum length.

Software Engineer Interview

Implement a linked list data structure in Java.

Software Engineer Interview

Design a financial instrument trading system. How would you ensure atomicity, consistency, and other key transactional properties in the system?

Software Engineer Interview

Given the head of a singly linked list, reverse the list and return the new head.

Software Engineer Interview

Given an array of points representing the elevations of a mountain (as a sequence of integers), and a specific position within this array, what is the highest point visible from that position?

Software Engineer Interview

Implement a queue using a stack.

Software Engineer Interview

Given a set of numbers [a, b, c, d, e] and a target value t, find the number of triplets in the set whose sum is at most t.

Software Engineer Interview

Given a CSV file as input, write a program to convert it to columnar data format.

Software Engineer Interview

Given a sentence containing three types of brackets (e.g., (), {}, []), determine if the brackets are properly balanced and in the correct order.

Software Developer Interview

Given a sorted array of integers and a target integer, can you find two numbers in the array such that their sum equals the target?

Software Engineer Interview

Describe an algorithm for calculating the square root of x.

Software Engineer Interview

Given a binary tree, write a function to determine if it is a valid binary search tree.

Developer Interview

Implement a hash table with collision handling using probing.

Software Engineer Interview

Design and implement a defaultdict class in Python using object-oriented design principles.

Software Engineer Interview

Given a matrix and the top-left and bottom-right coordinates of a submatrix, calculate the sum of the submatrix in (1) linear time and (2) constant time. You are allowed to preprocess the matrix.

Software Engineer Interview

Write code to implement a multi-reader/single-writer lock.

Software Engineer Interview

You have a worker and a gold bar. The worker will work for you for 15 days. He must have x/15 of the gold bar after the xth day of work. What is the minimum number of cuts you need to make to the gold bar to pay the worker accordingly?

Software Engineer Interview

Implement a function to compute the nth Fibonacci number using memoization.

Software Engineer Interview

Find the number of ways to get from one corner of a grid to the opposite corner, moving only down or right at each step.

Software Engineer Interview

How do you reverse a linked list?

Software Engineer Interview

Given an 8x8 chessboard, write code to calculate the number of ways to go from the top left corner to the bottom right corner, where you can move down, up, left, or right at each step.

Software Developer Interview

Explain the difference between Lasso and Ridge regression.

Quantitative Researcher Interview

Given three variables x, y, and z, where x is distributed as N(0,1), z is distributed as N(0, sigma^2), and y = x + z, with x independent of z: For a given observed value of y, what is the conditional distribution of x given y, i.e., p(x|y)?

Quantitative Researcher Interview

What is an O(n log n) sorting algorithm? Give an example and briefly explain how it works.

Quantitative Researcher Interview

Suppose you want to predict y and you have predictor vectors, each with known accuracy, variance, and sample length. How do you combine these predictors to minimize the residual standard error (RSE)?

Quantitative Researcher Interview

There are three random variables, X, Y, Z. The correlations between each pair of variables are the same, i.e., ρ = Corr(X,Y) = Corr(Y,Z) = Corr(Z,X). What is the tightest bound you can give for ρ? How about the general case for n random variables?

Quantitative Researcher Interview

What are the ways to construct a uniform distribution given only access to a fair coin?

Quantitative Researcher Interview

If 75 customers are randomly assigned to three equal-sized databases, and all partitions are equally likely, what is the probability that two randomly selected customers, Bob and Ben, are in the same database?

Quantitative Researcher Interview

Suppose you have several random variables that all have equal pairwise correlation. What is the possible range of this correlation value?

Quantitative Researcher Interview

What is the probability that X > Y, where X is distributed as N(0, 2) and Y is distributed as N(0, 1)?

Quantitative Researcher Interview

What is the expected number of samples drawn from a uniform distribution on [0,1] required so that their sum exceeds 1?

Quantitative Researcher Interview

What is the angle between the minute hand and the hour hand at 12:15?

Quantitative Researcher Interview

What happens to the optimal parameters of a linear regression if you feed in the same data twice? How about the R^2 and z-scores?

Quantitative Researcher Interview

For the task of classifying news articles into their subject type (e.g., finance, science, politics), what is a current state-of-the-art approach? Describe this approach in detail and discuss its latency. Additionally, suggest simpler, classical models that can achieve faster throughput while maintaining high accuracy for this task.

Quantitative Researcher Interview

1. In the context of the German tank problem, explain how to construct a statistical estimator for the maximum number of tanks produced, given a sample of observed serial numbers. 2. Calculate the mean and standard deviation of this estimator.

Quantitative Researcher Interview

Suppose there are 10 lions and a piece of meat. If any one of the lions eats the meat, that lion falls asleep. While asleep, any other lion can eat her and will also fall asleep. This process continues in the same way. What will happen at the beginning? Will any lion eat the meat?

Quantitative Researcher Interview

Given that the probability of getting heads on a coin flip is p, what is the expected number of flips required to get three heads in a row?

Quantitative Research Interview

What data structure is used to implement a dictionary in Python?

Quantitative Researcher Interview

Calculate the variance of x, denoted var(x), given that the data points are distributed uniformly on the surface of a 3D sphere.

Quantitative Researcher Interview

Write a program to find the square root of a number.

Quantitative Researcher Interview

Using stock data from the last five years, how can we build a model to predict the next day's price?

Quantitative Researcher Interview

You have r red balls and w white balls in a bag. If you keep drawing balls from the bag until only balls of a single color remain (i.e., you run out of one color), what is the probability you run out of white balls first? Express your answer in terms of r and w.

Quantitative Researcher Interview

What is correlation? What is covariance? Draw a graph where the correlation is equal to 1 and another where it is equal to -1.

Quantitative Researcher Interview

Create a class that implements a singly linked list data structure.

Quantitative Researcher Interview

Nine fair coins are tossed. What is the probability that an odd number of heads will land?

Quantitative Researcher Interview

Why is an L2 penalty added to linear regression models? Explain the effect of the L2 penalty on the solution.

Quantitative Researcher Interview

What is the expected number of coin flips required to see 2 heads in a series of fair coin tosses?

Quantitative Researcher Interview

The chance that a student passes a test is 10%. What is the probability that at least 50 out of 400 students pass the test? Choose the closest answer from: 5%, 10%, 15%, 20%, 25%.

Quantitative Researcher Interview

Why is regularization important, for example using ridge or lasso regression compared to ordinary least squares (OLS), even when the number of samples is greater than the number of parameters?

Quantitative Research Interview

What are virtual functions in C++? How are they used?

Quantitative Researcher Interview

What is the difference between regressing y on x and regressing x on y?

Quantitative Researcher Interview

Given a series of real numbers, what is the optimal point on the real line that minimizes the sum of the distances to all these numbers? Follow-up: What is the best algorithm to calculate the median of a sequence of numbers? Follow-up 2: Design an algorithm that can update the median of a sequence as new numbers are added.

Quantitative Researcher Interview

Given an array of numbers: 1, 5, 3, 2, 4, 2, 3, 1, 2, 3, 2, 4, return another array where duplicate numbers are grouped in consecutive positions, maintaining the order of their first appearance. For example, the result should be: 1, 1, 5, 3, 3, 3, 2, 2, 2, 2, 4, 4.

Quantitative Researcher Interview

Given an array of numbers, find three numbers such that their product is the maximum possible.

Quantitative Researcher Interview

Write a program to output all the prime numbers smaller than a given integer N.

Quantitative Researcher Interview

Find the minimum value of the function f(x) = x^x for real x > 0.

Quantitative Researcher Interview

You have two eggs and a 100-story building. If an egg is dropped from a floor below floor X, it will not break; if dropped from floor X or higher, it will always break. What strategy would you use to determine the exact value of X with the minimum number of drops in the worst-case scenario?

Quantitative Researcher Interview

How will the probability of an option ending up in the money change if the volatility of the underlying stock increases?

Quantitative Researcher Interview

What is the difference between Gaussian Naive Bayes (GNB) and logistic regression? Under what circumstances would you choose one over the other?

Quantitative Researcher Interview

Given that log X is normally distributed with mean 0 and variance 1, compute the expected value of X.

Quantitative Researcher Interview

How do you estimate the square root of 5?

Quantitative Researcher Interview

You have two strings, each of which takes exactly one hour to burn completely when lit at one end. The rate of burning is non-uniform and differs between the strings. Using only these two strings and a way to light them, how can you measure 45 minutes?

Quantitative Researcher Interview

Write a function to calculate Fibonacci numbers using recursion.

Quantitative Analyst Interview

Suppose you roll two fair six-sided dice. What is the probability that the sum of the two dice is equal to 8?

Quantitative Analyst Interview

How would you run a Monte Carlo simulation in Python to predict the probability of obtaining 4 out of 6 coins as heads, without using for loops?

Quantitative Analyst Interview

What is the expected value of a fair six-sided die?

Quantitative Analyst Interview

1. You have two fair dice. What is the probability that their sum is a prime number? 2. Three people play a game where each flips a coin. The last person with a different result wins. What is the expected number of flips? 3. What is 17 multiplied by 23? 4. What is the square root of 2907 to two decimal places? 5. If the S&P moves 0.3% and AAPL's beta is 1.2, what is AAPL's expected move? 6. Write a function to find arbitrage opportunities in a currency exchange matrix. 7. Given real-time trade data, design an algorithm to detect wash trading. 8. Implement a fast moving median calculator for streaming data. 9. How would you arbitrage S&P 500 futures versus the underlying basket? 10. What happens to options prices during earnings announcements? 11. Design a mean-reversion strategy for ETF pairs trading. 12. 100 people wear either red or blue hats. Each person can see others' hats but not their own. They must all simultaneously guess their own hat color. They can plan beforehand. How do they maximize group survival?

Quantitative Analyst Interview

What are the assumptions of linear regression?

Quantitative Analyst Interview

How would you design a machine learning algorithm that uses Twitter data to generate a trading strategy?

Quantitative Analyst Interview

What is the probability of finding two consecutive ones in a randomly generated binary string of length n, where each bit is equally likely to be 0 or 1?

Trader Interview

What is the probability of picking two diamonds and one card that is not a diamond from a standard deck of 52 cards?

Trader Interview

Suppose I have two pancakes: one with both sides burnt, and one with one burnt side and one unburnt side. I randomly choose one pancake, then randomly look at one of its sides. If the side is burnt, what is the probability that I picked the pancake with both sides burnt?

Trader Interview

What is the expected value of a fair six-sided die?

Trader Interview

What is the probability of getting a three-of-a-kind in Texas Hold'em poker?

Trader Interview

Implement a class for the game Connect 4.

Software Engineer Intern Interview

Given an array v, find the number of pairs of indices (i, j) such that v[i] = v[j] = v[i+1] + v[i+2] + ... + v[j-1]. Your solution should have time complexity O(n log n).

Software Engineer Intern Interview

How would you solve a scheduling issue given a directed acyclic graph (DAG) of tasks and k computers? What if k is one or infinity?

Software Engineer Intern Interview

Design an algorithm to randomly select people in a city such that the probability of selecting someone from a given district is proportional to the district's population size. Additionally, discuss how you would test your implementation and optimize its runtime, considering repeated calls to the selection function with an acceptable error margin.

Software Developer Intern Interview

If a Binary Search Tree was implemented incorrectly such that a node has a left child that is not smaller than itself, how would you detect this violation? What is the most efficient method to identify this issue?

Software Developer Intern Interview

There are three random variables, X, Y, and Z. The three correlations between the variables are equal, i.e., ρ = Corr(X, Y) = Corr(Y, Z) = Corr(Z, X). What is the tightest possible bound you can give for ρ? How about the general case for n random variables?

Quant Researcher Interview

What is the difference between LASSO and ridge regression?

Quant Research Interview

Describe linear regression in detail.

Quant Researcher Interview

Given an n x n chessboard, return the minimum number of knight moves required to get from point (a, b) to point (c, d).

Software Engineering Intern Interview

Return a minimum-length list of perfect squares whose sum equals a given value k.

Software Engineering Intern Interview

Reverse a linked list in place.

Software Engineering Intern Interview

Implement a balanced binary search tree from scratch, explain its time complexity, and optimize it to efficiently handle duplicate values.

Software Engineer Interview

Given an array representing user events, find the length of the longest contiguous subarray where the most frequent user in that subarray has the same number of events as the least frequent user in the entire event log. Write an efficient function to solve this problem.

Software Engineer Interview

How can a deadlock occur?

Software Engineer Interview

Design a financial instrument trading system. How would you ensure atomicity, consistency, and other properties in the system?

Software Engineer Interview

Given the head of a singly linked list, reverse the list and return the new head.

Software Engineer Interview

Given an array representing heights of points on a mountain and a starting position, what is the highest point that is visible from that position?

Software Engineer Interview

Implement a queue using a stack.

Software Engineer Interview

100

Given a set of numbers [a, b, c, d, e] and a target t, find the number of triplets from the set whose sum is less than or equal to t.

Software Engineer Interview

101

Given a sentence containing three types of brackets—parentheses (), square brackets [], and curly braces {}—determine if the brackets are properly matched and ordered.

Software Developer Interview

102

Given a sorted array of integers, can you find two numbers such that their sum equals a target integer?

Software Engineer Interview

103

Describe an algorithm for calculating the square root of x.

Software Engineer Interview

104

Implement a hashtable data structure that handles hash collisions using probing.

Software Engineer Interview

105

In Python, use object-oriented design to implement a defaultdict from scratch.

Software Engineer Interview

106

Given a matrix and the coordinates of the top-left and bottom-right corners, calculate the sum of the elements within the specified submatrix in (1) linear time and (2) constant time. You are allowed to preprocess the matrix.

Software Engineer Interview

107

Write the code for a multi-reader/single-writer lock.

Software Engineer Interview

108

You have a worker and a gold bar. The worker will work for you for 15 days. He must have x/15 of the gold bar after the xth day of work. What is the minimum number of cuts needed to divide the gold bar so that you can pay the worker accordingly?

Software Engineer Interview

109

Implement a function to compute the nth Fibonacci number using memoization.

Software Engineer Interview

110

Find the number of ways to get from one corner of a grid to the opposite corner, moving only down or right at each step.

Software Engineer Interview

111

How do you reverse a linked list?

Software Engineer Interview

112

Write a program to schedule a stream of pizza delivery orders.

Software Engineer Interview

113

Given an 8x8 chessboard, write code to calculate the number of ways to go from the top left corner to the bottom right corner, given that you can move down, up, left, and right.

Software Developer Interview

114

Calculate the Sharpe ratio of a 60/40 coin toss where each toss results in a gain of $1 or a loss of $1, respectively.

Internship Interview

115

What is a probability density function?

Internship Interview

116

Explain how to use a binomial tree to determine the price of a European call option.

Internship Interview

117

How do you reverse a string using Python?

Internship Interview

118

How can you generate a probability of 1/4, 1/3, or 1/π using only coin flips?

Quantitative Research Analyst Interview

119

Suppose you have three random variables. The correlation between A and B is 0.6, and the correlation between A and C is 0.8. What is the possible range of the correlation between B and C?

Quantitative Research Analyst Interview

120

You are given an array of integers, which may include both negative and positive values, and another integer S. Find three different integers in the array whose sum is closest to S. If multiple solutions exist, any is acceptable. Is there an algorithm to find the three integers in O(n^2) time?

Quantitative Research Analyst Interview

121

If you have a dataset with many features, what methods can you use to reduce its dimensionality?

Quant Research Intern Interview

122

How do you derive the Black-Scholes formula for pricing European options?

Quant Research Intern Interview

123

1. Describe your past research experience related to reinforcement learning and model explainability. 2. Explain the bias-variance trade-off in random forests. 3. What is the closed-form solution to linear regression, with and without L2 regularization?

Quant Research Intern Interview

124

Analyze the variance of a momentum trading strategy.

Quant Research Intern Interview

125

Sample points randomly from a unit sphere. If you generate independent standard normal variables X1, X2, X3 for the three coordinates and normalize by dividing by sqrt(X1^2 + X2^2 + X3^2), then (X1, X2, X3) / sqrt(X1^2 + X2^2 + X3^2) is a uniform random point on the sphere. What is the variance of each coordinate of the resulting uniform distribution on the sphere?

Quant Research Intern Interview

126

Given the numbers 1 to 1000 arranged around a circle in order, start at 1 and delete every alternate number (i.e., remove 1, then 3, then 5, and so on). Continue this process of deleting every other remaining number, wrapping around the circle as necessary, until only one number remains. Determine this final remaining number.

Quant Research Intern Interview

127

How do you generate a uniform distribution over a disk given access to two independent uniform random variables?

Quantitative Trader Interview

128

What is the expected value of the cube of a fair six-sided die roll?

Quantitative Trader Interview

129

Given a matrix of size N by T, where the entry at the i-th row and j-th column represents the price of buying i shares of a stock at time j, determine the minimum total cost of purchasing N shares by the end of time T.

Quantitative Trader Interview

130

You are given two ropes that, when lit, each burn for exactly one hour, but not necessarily at a constant rate. Which one of the following times CANNOT be measured using these ropes? a) 50 minutes b) 30 minutes c) 25 minutes d) 35 minutes.

Quantitative Trader Interview

131

Implement the native map function for arrays.

Senior Software Engineer Interview

132

In C++, would you prefer to store pointers or references in a vector? Explain your reasoning.

Senior Software Engineer Interview

133

In C++ templates, in what cases is the keyword 'typename' used?

Senior Software Engineer Interview

134

How do you implement a hash map in C#?

Senior Software Developer Interview

135

What is the expected value when rolling two fair six-sided dice?

Summer Intern Interview

136

Write code to compute x raised to the power n (x^n) in O(log n) time.

Summer Intern Interview

137

Implement the k-nearest neighbor algorithm.

Quantitative Developer Interview

138

Given a vector of edges (e.g., A-B, A-C, C-D), write a function to determine if the resulting graph is a tree. If it is a tree, return its root.

Quantitative Developer Interview

139

Implement a class for the game Connect Four.

Software Engineer Intern Interview

140

Find the number of pairs of indices (i, j) in an array v such that v[i] = v[j] = v[i+1] + v[i+2] + ... + v[j-1]. Design an algorithm with time complexity O(n log n).

Software Engineer Intern Interview

141

How would you solve a scheduling issue given a directed acyclic graph (DAG) of tasks and k computers? What if k is one or infinity?

Software Engineer Intern Interview

142

If a Binary Search Tree was implemented incorrectly, such that a number to the left of a node is not smaller as it should be, how would you detect this mistake? What is the most efficient way to identify such an error?

Software Developer Intern Interview

143

Two people plan to meet at a train station. Each arrives at a random time between 4:00 pm and 5:00 pm and will wait 10 minutes for the other. What is the probability that they will meet?

Quantitative Researcher Intern Interview

144

What is implied volatility? What is the probability distribution of X + Y, where X and Y are independent and identically distributed uniform random variables?

Quantitative Researcher (Intern) Interview

145

Given a list of stock prices for consecutive days, determine the optimal strategy to maximize profit from buying and selling stocks. You may complete as many transactions as you like (i.e., buy and sell multiple times), but you must sell the stock before you buy again.

Quantitative Researcher (Intern) Interview

146

1. Given a triangle defined by its vertices, determine whether a given point is inside the triangle. 2. Given a list of times, find the number of pairs of times that add up to 60.

Software Engineering Interview

147

Explain why the volatility smile exists.

Trading Interview

148

You are tossing a fair coin. What is the expected number of tosses until you get two heads in a row?

Trading Interview

149

If 0.1% of people have a certain disease, and you have a test that is correct 99% of the time, what is the probability that someone who tests positive actually has the disease?

Trading Intern Interview

150

What is the expected value of the sum when rolling two fair six-sided dice?

Quant Trading Intern Interview

151

Given a biased coin (with probability p of landing heads), what is the probability of flipping two heads (HH) in two consecutive tosses?

Quant Trading Intern Interview

152

Describe how the random forest algorithm works.

Quant Trading Intern Interview

Quant Interview Questions

What is covariance?

Build a linked list in Java.

Implement a balanced binary search tree from scratch and explain its time complexity. Additionally, optimize the tree to efficiently handle duplicate values.

Implement a linked list data structure in Java.

Design a financial instrument trading system. How would you ensure atomicity, consistency, and other key transactional properties in the system?

Given the head of a singly linked list, reverse the list and return the new head.

Given an array of points representing the elevations of a mountain (as a sequence of integers), and a specific position within this array, what is the highest point visible from that position?

Implement a queue using a stack.

Given a set of numbers [a, b, c, d, e] and a target value t, find the number of triplets in the set whose sum is at most t.

Given a CSV file as input, write a program to convert it to columnar data format.

Given a sentence containing three types of brackets (e.g., (), {}, []), determine if the brackets are properly balanced and in the correct order.

Given a sorted array of integers and a target integer, can you find two numbers in the array such that their sum equals the target?

Describe an algorithm for calculating the square root of x.

Given a binary tree, write a function to determine if it is a valid binary search tree.

Implement a hash table with collision handling using probing.

Design and implement a defaultdict class in Python using object-oriented design principles.

Given a matrix and the top-left and bottom-right coordinates of a submatrix, calculate the sum of the submatrix in (1) linear time and (2) constant time. You are allowed to preprocess the matrix.

Write code to implement a multi-reader/single-writer lock.

You have a worker and a gold bar. The worker will work for you for 15 days. He must have x/15 of the gold bar after the xth day of work. What is the minimum number of cuts you need to make to the gold bar to pay the worker accordingly?

Implement a function to compute the nth Fibonacci number using memoization.

Find the number of ways to get from one corner of a grid to the opposite corner, moving only down or right at each step.

How do you reverse a linked list?

Given an 8x8 chessboard, write code to calculate the number of ways to go from the top left corner to the bottom right corner, where you can move down, up, left, or right at each step.

Explain the difference between Lasso and Ridge regression.

Given three variables x, y, and z, where x is distributed as N(0,1), z is distributed as N(0, sigma^2), and y = x + z, with x independent of z: For a given observed value of y, what is the conditional distribution of x given y, i.e., p(x|y)?

What is an O(n log n) sorting algorithm? Give an example and briefly explain how it works.

Suppose you want to predict y and you have predictor vectors, each with known accuracy, variance, and sample length. How do you combine these predictors to minimize the residual standard error (RSE)?

There are three random variables, X, Y, Z. The correlations between each pair of variables are the same, i.e., ρ = Corr(X,Y) = Corr(Y,Z) = Corr(Z,X). What is the tightest bound you can give for ρ? How about the general case for n random variables?

What are the ways to construct a uniform distribution given only access to a fair coin?

If 75 customers are randomly assigned to three equal-sized databases, and all partitions are equally likely, what is the probability that two randomly selected customers, Bob and Ben, are in the same database?

Suppose you have several random variables that all have equal pairwise correlation. What is the possible range of this correlation value?

What is the probability that X > Y, where X is distributed as N(0, 2) and Y is distributed as N(0, 1)?

What is the expected number of samples drawn from a uniform distribution on [0,1] required so that their sum exceeds 1?

What is the angle between the minute hand and the hour hand at 12:15?

What happens to the optimal parameters of a linear regression if you feed in the same data twice? How about the R^2 and z-scores?

1. In the context of the German tank problem, explain how to construct a statistical estimator for the maximum number of tanks produced, given a sample of observed serial numbers. 2. Calculate the mean and standard deviation of this estimator.

Suppose there are 10 lions and a piece of meat. If any one of the lions eats the meat, that lion falls asleep. While asleep, any other lion can eat her and will also fall asleep. This process continues in the same way. What will happen at the beginning? Will any lion eat the meat?

Given that the probability of getting heads on a coin flip is p, what is the expected number of flips required to get three heads in a row?

What data structure is used to implement a dictionary in Python?

Calculate the variance of x, denoted var(x), given that the data points are distributed uniformly on the surface of a 3D sphere.

Write a program to find the square root of a number.

Using stock data from the last five years, how can we build a model to predict the next day's price?

You have r red balls and w white balls in a bag. If you keep drawing balls from the bag until only balls of a single color remain (i.e., you run out of one color), what is the probability you run out of white balls first? Express your answer in terms of r and w.

What is correlation? What is covariance? Draw a graph where the correlation is equal to 1 and another where it is equal to -1.

Create a class that implements a singly linked list data structure.

Nine fair coins are tossed. What is the probability that an odd number of heads will land?

Why is an L2 penalty added to linear regression models? Explain the effect of the L2 penalty on the solution.

What is the expected number of coin flips required to see 2 heads in a series of fair coin tosses?

The chance that a student passes a test is 10%. What is the probability that at least 50 out of 400 students pass the test? Choose the closest answer from: 5%, 10%, 15%, 20%, 25%.

Why is regularization important, for example using ridge or lasso regression compared to ordinary least squares (OLS), even when the number of samples is greater than the number of parameters?

What are virtual functions in C++? How are they used?

What is the difference between regressing y on x and regressing x on y?

Given an array of numbers: 1, 5, 3, 2, 4, 2, 3, 1, 2, 3, 2, 4, return another array where duplicate numbers are grouped in consecutive positions, maintaining the order of their first appearance. For example, the result should be: 1, 1, 5, 3, 3, 3, 2, 2, 2, 2, 4, 4.

Given an array of numbers, find three numbers such that their product is the maximum possible.

Write a program to output all the prime numbers smaller than a given integer N.

Find the minimum value of the function f(x) = x^x for real x > 0.

You have two eggs and a 100-story building. If an egg is dropped from a floor below floor X, it will not break; if dropped from floor X or higher, it will always break. What strategy would you use to determine the exact value of X with the minimum number of drops in the worst-case scenario?

How will the probability of an option ending up in the money change if the volatility of the underlying stock increases?

What is the difference between Gaussian Naive Bayes (GNB) and logistic regression? Under what circumstances would you choose one over the other?

Given that log X is normally distributed with mean 0 and variance 1, compute the expected value of X.

How do you estimate the square root of 5?

You have two strings, each of which takes exactly one hour to burn completely when lit at one end. The rate of burning is non-uniform and differs between the strings. Using only these two strings and a way to light them, how can you measure 45 minutes?

Write a function to calculate Fibonacci numbers using recursion.

Suppose you roll two fair six-sided dice. What is the probability that the sum of the two dice is equal to 8?

How would you run a Monte Carlo simulation in Python to predict the probability of obtaining 4 out of 6 coins as heads, without using for loops?

What is the expected value of a fair six-sided die?

What are the assumptions of linear regression?

How would you design a machine learning algorithm that uses Twitter data to generate a trading strategy?

Write a Python function to calculate the variance of a given list of numbers.

How can you detect a loop in a linked list?

Write a function to traverse a binary tree without using recursion.

How would you construct an options position that is long vega and short gamma?

What is the probability of finding two consecutive ones in a randomly generated binary string of length n, where each bit is equally likely to be 0 or 1?

What is the probability of picking two diamonds and one card that is not a diamond from a standard deck of 52 cards?

Suppose I have two pancakes: one with both sides burnt, and one with one burnt side and one unburnt side. I randomly choose one pancake, then randomly look at one of its sides. If the side is burnt, what is the probability that I picked the pancake with both sides burnt?

What is the expected value of a fair six-sided die?

What is the probability of getting a three-of-a-kind in Texas Hold'em poker?

Implement a class for the game Connect 4.

Given an array v, find the number of pairs of indices (i, j) such that v[i] = v[j] = v[i+1] + v[i+2] + ... + v[j-1]. Your solution should have time complexity O(n log n).