Exercise: Bernoulli Sums and the de Moivre-Laplace Approximation

Prerequisites: Central Limit Theorem, Bernoulli and Binomial Distributions

Problem

A hedge fund runs a directional strategy that picks the right side of the market on each of $n$ independent trading days with probability $p = 0.55$. Let $S_n$ be the number of winning days out of $n$.

1. Compute $\mathbb{E}[S_n]$ and $\operatorname{Var}(S_n)$ exactly. Apply the CLT to approximate $S_n$ as a Gaussian. What are the approximating mean and variance for $n = 100$?

2. Using the Gaussian approximation, estimate $\mathbb{P}(S_{100} \ge 60)$ — the probability of at least 60 winning days out of 100.

3. Apply the continuity correction $\mathbb{P}(S_{100} \ge 60) \approx 1 - \Phi\!\left(\frac{59.5 - np}{\sqrt{np(1-p)}}\right)$ and recompute. Compare to the answer from part 2.
4. Compute the same probability exactly using the binomial CDF (use Python's scipy.stats.binom or equivalent). Which Gaussian approximation — with or without continuity correction — matches the exact answer better?

Hint