The normal distribution

The video below shows an example of a Galton board or a Quincunx board, which demonstrates how the normal distribution arises from the sum of many independent random variables.

This video demonstrates a device known as a Galton Board (also called a quincunx or bean machine). It is a desktop probability machine that visualizes abstract statistical concepts.

Here is a breakdown of what is happening:

This machine, invented by Sir Francis Galton in the 19th century, illustrates the Central Limit Theorem. For independent, identically distributed steps with finite, nonzero variance, the standardized sum approaches a standard normal distribution as the number of steps grows. A physical board is an approximation to this model: collisions and friction can affect the beads’ paths.

Simulating a Galton board

Follow one ball through the pegs, then drop a whole sample. In this idealized board, every bounce is an independent choice: right with probability \(p\), left with probability \(1-p\). The landing bin \(K\) counts the right bounces, so after \(n\) rows,

\[ \begin{gathered} K \sim \operatorname{Binomial}(n,p), \\ \mu = np, \qquad \sigma = \sqrt{np(1-p)}. \end{gathered} \]

The blue bars show the balls that have landed. Compare them with the exact binomial probabilities and the normal approximation, both calculated from your current settings.

Try these three experiments:

  1. More balls, same board: keep 16 rows and \(p=0.5\), then compare 100 with 5,000 balls. The observed shares become more stable around the binomial probabilities: this is the Law of Large Numbers.
  2. More rows: compare 4 with 30 rows at \(p=0.5\). The binomial shape becomes closer to a normal curve as the number of independent steps increases, illustrating the Central Limit Theorem.
  3. Bias, drift, and variability: change \(p\) to 70%. The mean landing bin moves to the right, and the distribution becomes narrower than at 50%. Then try 100%: every bounce goes right, so all balls land in the last bin, \(K=n\). There is no randomness left.

Changing the right-bounce probability introduces drift in this random walk, but also changes its variance. If each right move is \(+1\) and each left move is \(-1\), the expected step is \(2p-1\) and its variance is \(4p(1-p)\). After \(n\) rows, the displacement is \(2K-n\), with expected value \(n(2p-1)\). Variability is greatest at \(p=50\%\) and falls to zero at \(p=0\%\) or \(100\%\).

This is a limited analogy for drift in an asset-return model. In a Brownian-motion model, drift and volatility are separate parameters: increasing drift can shift the distribution without changing its spread. The board’s probability slider changes both, so it is not a pure drift control.

Visualizing the normal density

Below you can find a plot of two density functions from a Normal distribution. The blue curve shows the standard Normal(0, 1), while the red curve lets you adjust the mean and standard deviation to see how the shape shifts.

The relationship between a general normal variable \(X\) and the standard normal variable \(Z\) is given by the Z-score formula: \[ Z = \frac{X - \mu}{\sigma} \] This standardization allows us to compare different normal distributions and is fundamental for calculations like Value-at-Risk (VaR).

Change those parameters using the sliders below and see how this affects the density function (area in red).

For more on the Normal distribution refer to:

The normal density for \(X \sim \mathcal{N}(\mu, \sigma^2)\) is

\[ f_X(x) = \frac{1}{\sigma \sqrt{2\pi}}\exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right). \]

NoteFinancial Context: Returns vs. Prices

In finance, we often assume that the log-returns of an asset are normally distributed: \[ \ln(S_t/S_{t-1}) \sim \mathcal{N}(\mu, \sigma^2) \] This implies that the asset prices themselves follow a Log-Normal Distribution.

  • Limited Liability: While returns can be negative, asset prices cannot be negative. This is a key reason we model prices as Log-Normal (domain \(0\) to \(\infty\)) rather than Normal (domain \(-\infty\) to \(\infty\)).

The Cumulative Distribution Function (CDF) of the standard normal distribution, often denoted as \(N(x)\) or \(\Phi(x)\), represents the probability that a variable takes a value less than or equal to \(x\) (the area under the curve to the left). This plays a critical role in the Black-Scholes-Merton option pricing model (specifically in the \(N(d_1)\) and \(N(d_2)\) terms).

Critique: Real-world financial data often deviates from normality:

  • “Fat Tails” (Excess Kurtosis): Extreme events (crashes or booms) happen more frequently than predicted.
  • Skewness: Markets often exhibit asymmetry (e.g., equity markets frequently have negative skewness, implying higher crash risk).
NoteReference

This page accompanies Chapters 14 and 15 of Hull (2022).

References

Hull, John. 2022. Options, Futures, and Other Derivatives. 11th ed. Pearson.