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Solve Probability and Statistics Questions

SquarepointQuantitative Trader · Technical Screen
MediumQuantitative TraderTechnical ScreenMachine Learning

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This problem tests your understanding of OLS regression assumptions, the asymptotic behavior of sample estimators, and expected value optimization under probabilistic constraints.

Part 1: Linear Regression Inference

You are given a dataset of n observations (x_i, y_i), where y_i = beta_0 + beta_1*x_i + epsilon_i. Assume the errors epsilon_i are i.i.d. with mean 0 and variance sigma^2, and are independent of x_i.

  1. Derive the OLS estimator for beta_1.
  2. Explain what happens to the variance of the estimator beta_1_hat as n approaches infinity, assuming the variance of x is finite and non-zero.

Part 2: Asymptotic Probability

You have a sequence of independent random variables X_1, X_2, ..., X_n, where each X_i follows a Bernoulli distribution with success probability p = 0.6. Let S_n = sum(X_i) / n.

  1. State the Law of Large Numbers (LLN) and the Central Limit Theorem (CLT) in the context of S_n.
  2. For n = 100, approximate the probability that S_n > 0.65 using the Normal distribution.

Part 3: Strategy Optimization

You are playing a game where you have a bankroll of $10,000. Each round, you can bet an amount b on an event with a win probability p = 0.55. If you win, you double your bet; if you lose, you lose the bet.

  1. Define the Kelly Criterion and calculate the optimal fraction of your bankroll to wager per round to maximize the long-term growth rate of your capital.

Constraints

  • Use the approximation N(0, 1) for the distribution of the sample mean.
  • Assume you have access to standard statistical tables if necessary.
  • Express your answers as closed-form expressions or numerical values rounded to four decimal places.

Hints

Hint 1

For OLS, recall that the estimator minimizes the sum of squared residuals; take the derivative with respect to beta_1 and set to zero.

Hint 2

For the Bernoulli approximation, use the variance formula Var(S_n) = p(1-p)/n to normalize the variable into a Z-score.

Hint 3

The Kelly Criterion formula for a binary outcome is f* = (bp - q) / b, where b is the odds received (here, 1:1) and q = 1-p.

Follow-up questions

  • How would the OLS estimator change if we introduced heteroscedasticity (where the variance of epsilon_i depends on x_i)?
  • What are the practical limitations of using the Kelly Criterion in a real-world trading environment?
  • If the X_i variables were correlated rather than independent, how would the Law of Large Numbers be affected?