TL;DR

The distribution of the number of successes in n independent Bernoulli trials: \( P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \). Mean is np, variance is np(1-p).

By Valenke Exam Prep Team·Last updated 2026-06-03

Binomial Distribution

The distribution of the number of successes in n independent Bernoulli trials: \( P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \). Mean is np, variance is np(1-p).

Why it matters for interviews

Foundation for discrete probability, the binomial tree pricing model, and counting problems. Interview questions frequently involve computing binomial probabilities, deriving moments, and connecting to the normal via CLT.

Definition and Mathematical Foundation

The distribution of the number of successes in n independent Bernoulli trials: \( P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \). Mean is np, variance is np(1-p).

Application in Quantitative Finance

Related Terms

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Frequently Asked Questions

When does the binomial approximate the Poisson?▼

When n is large and p is small, with \( np = \lambda \) moderate: \( \binom{n}{k}p^k(1-p)^{n-k} \approx e^{-\lambda}\lambda^k/k! \). Rule of thumb: n > 20, p < 0.05.

When does the binomial approximate the normal?▼

When both np and n(1-p) exceed 5. The normal approximation \( X \approx N(np, np(1-p)) \) with continuity correction is accurate. This is a direct application of the CLT.

How does the binomial distribution connect to the binomial tree?▼

After n periods in the CRR model, the terminal stock price has been multiplied by \( u^k d^{n-k} \) with probability \( \binom{n}{k} p^k (1-p)^{n-k} \). The option price is a binomial expectation.