Polson–Sokolov Lab · Normal distribution and fat tails

Normal Tails Lab

Shade a tail of the normal curve and read its probability. Standardize the cutoff to z, then turn the probability into "once in how many years". Load a historic market crash and see how a fat-tailed Student-t changes the answer.

← All labs

Based on Chapter 1, Probability and Uncertainty, of Polson and Sokolov, Bayes, Artificial Intelligence, and Deep Learning (vsokolov.org/html/_book).

Density

Shaded probability Normal N(μ, σ²) Student-t, same mean and SD Cutoff

Tail probability on a log scale

Standardize, step by step
    Extreme market events

    A 5-sigma month and a 7-sigma day

    Before October 1987, S&P 500 monthly returns had \(\mu = 1.2\%\) and \(\sigma = 4.3\%\). The crash month was −21.76%, so \(z = (-21.76 - 1.2) / 4.3 = -5.34\). Under the normal that is about 5 in 100 million, or once in 1.8 million years of months. NVIDIA's daily returns since 2023 have mean 0.38% and SD 3.3%; the +24.7% day on May 25, 2023 is \(z \approx 7.4\).

    1. Only intervals have probability

      For a continuous random variable, \(P(X = a) = 0\). Probabilities are areas under the density, \(P(a < X < b) = \int f(x)\,dx\) from \(a\) to \(b\). The shaded region is that integral.

    2. Standardize to use one table

      \(Z = (X - \mu) / \sigma\) is \(N(0, 1)\) for any normal \(X\). So \(P(X < a) = \Phi((a - \mu)/\sigma)\), and every normal question becomes a question about \(z\). In R this is pnorm(a, mean, sd); in Excel NORMDIST(a, mean, sd, TRUE).

    3. Turn a probability into a waiting time

      If one draw is a month, a tail probability \(p\) means about \(12p\) such months per year, so the event happens once every \(1/(12p)\) years on average. For daily returns, use 252 trading days.

    4. The normal understates extremes

      The log-scale chart shows why. Normal tail probabilities fall like \(e^{-z^2/2}\), so each extra sigma divides the chance by a growing factor. A Student-t tail falls like a power of \(z\). At 5 to 7 SDs the gap is many orders of magnitude.

    Normal density and standardization
    \(\displaystyle f(x) = \frac{\exp\big(-(x - \mu)^2 / (2\sigma^2)\big)}{\sqrt{2\pi}\,\sigma} \)
    \(\displaystyle Z = \frac{X - \mu}{\sigma} \sim N(0, 1) \)
    Common quantiles, checked in this page
    Upper tailzUse
    Historic events under the normal
    EventzNormal p

    Takeaway for students. The normal is a useful approximation for typical days and months, and it is badly wrong about the extremes that matter for risk. Other examples: the COVID crash (S&P 500 down 34% in 33 days, March 2020), the 2022 rate shock (60/40 portfolios lost 16%), VIX at 65 in August 2024, and NVIDIA's DeepSeek day. There are no \(\mu\) and \(\sigma\) here for the first three, so they are not presets here. In an A/B test the same lesson reads: be careful with p-values when the metric is not normal.

    Try this in class

    1. Load "NVIDIA, lose ≥ 5% in a day". Check \(z = -1.63\) and \(P \approx 5\%\). Ask how many such days a year that implies (about 13).
    2. Load the 1987 crash. Read the normal's waiting time, then the Student-t's with \(\nu = 4\). Ask which one matches a market that has had several crashes in a century.
    3. Load the NVIDIA 7-sigma day and move \(\nu\) from 30 down to 3. Watch the purple tail line on the log chart lift by many orders of magnitude while the density barely changes shape.
    4. Press "\(\mu \pm 2\sigma\)" and check the empirical rule: 95.4% for the normal. With the t at \(\nu = 4\) the band holds about the same (95.3%), yet the chance of a move beyond \(3\sigma\) is 1.3% instead of 0.27%. Fat tails hide out past the middle.
    5. Load the 5-sigma Higgs standard. Confirm 1 in 3.5 million, the physics discovery threshold, and compare with how often 5-sigma market moves really happen.