Bernoulli and Binomial Distributions

Motivation: why this matters in quant finance

A Bernoulli variable models one binary event: default or no default, up move or down move, VaR exception or no exception, profitable trade or losing trade. A binomial variable counts how many such events occur across repeated independent trials.

That makes the Bernoulli/binomial pair the discrete skeleton behind option trees, default counts, exception backtests, and binary classifiers. ISL's credit-default examples are Bernoulli models with probabilities that depend on predictors. A binomial tree is a repeated Bernoulli model under a risk-neutral probability.

Definition

A Bernoulli random variable

X\sim\operatorname{Bern}(p)

satisfies

\mathbb{P}(X=1)=p, \qquad \mathbb{P}(X=0)=1-p.

If $X_1,\ldots,X_n$ are independent $\operatorname{Bern}(p)$ variables, then

S_n=\sum_{i=1}^{n}X_i\sim\operatorname{Bin}(n,p),

with PMF

\mathbb{P}(S_n=k)=\binom{n}{k}p^k(1-p)^{n-k}, \qquad k=0,1,\ldots,n.

Key Properties

For $X\sim\operatorname{Bern}(p)$ ,

\mathbb{E}[X]=p, \qquad \operatorname{Var}(X)=p(1-p).

For $S_n\sim\operatorname{Bin}(n,p)$ ,

\mathbb{E}[S_n]=np, \qquad \operatorname{Var}(S_n)=np(1-p).

If independent binomials share the same $p$ , their counts add:

\operatorname{Bin}(n_1,p)+\operatorname{Bin}(n_2,p)\sim\operatorname{Bin}(n_1+n_2,p).

For large $n$ with $p$ away from the edges,

\frac{S_n-np}{\sqrt{np(1-p)}}\xrightarrow{d}\mathcal{N}(0,1).

For rare events with $np\to\lambda$ ,

\operatorname{Bin}(n,p)\xrightarrow{d}\operatorname{Pois}(\lambda).

In Quant Finance

In a one-period binomial option tree, the stock moves to $S_0u$ or $S_0d$ . The risk-neutral probability is

\widetilde{p}=\frac{e^{r\Delta t}-d}{u-d}.

After $n$ steps, the number of up moves is $\operatorname{Bin}(n,\widetilde{p})$ . The option price is a discounted expectation under $\widetilde{p}$ , not a forecast under the real-world probability.

For VaR backtesting, each day is an exception indicator. Under a correct 99% VaR model, the exception probability is $p=0.01$ , so the annual exception count is modelled as $\operatorname{Bin}(250,0.01)$ before dependence and regime effects are considered.

Worked Example: Exception Count

If $X\sim\operatorname{Bin}(250,0.01)$ , then $\mathbb{E}[X]=2.5$ . Observing 8 exceptions requires checking

\mathbb{P}(X\geq8)=1-\sum_{k=0}^{7}\binom{250}{k}(0.01)^k(0.99)^{250-k}.

A high exception count may mean the VaR model is poor, or that the independence and constant-probability assumptions in the binomial backtest are poor.

Common Confusions and Pitfalls

The tree probability is a belief that the stock goes up. In pricing, it is a risk-neutral probability.

Any binary count is binomial. Only if trials are independent and share the same success probability.

A high win rate proves a strategy works. A Bernoulli count ignores payoff size, costs, dependence, and selection bias.

Where This Goes Next

Poisson Distribution: the rare-event limit of binomial counts.
Normal Distribution: the large-sample approximation for non-rare binomial counts.
Binomial Tree Model: repeats Bernoulli up/down moves to price derivatives.

References

Bertsekas and Tsitsiklis, Introduction to Probability, 2nd ed., Ch. 1 Sec. 1.5-1.6, Ch. 2 Sec. 2.2, Sec. 2.4, Sec. 2.7, and Ch. 4 Sec. 4.4.
Mosteller, Fifty Challenging Problems in Probability, Problems 27-28 and 44, for binomial sampling and repeated-play decision framing.
James, Witten, Hastie, and Tibshirani, An Introduction to Statistical Learning, 2nd ed., Ch. 4, for binary response modelling and the credit-default example.