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.

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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 μ = 1.2% and σ = 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 ≈ 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) = ∫ f(x) dx from a to b. The shaded region is that integral.

    2. Standardize to use one table

      Z = (X − μ) / σ is N(0, 1) for any normal X. So P(X < a) = Φ((a − μ)/σ), 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−z²/2, 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
    f(x) = exp(−(x − μ)² / (2σ²)) / (√(2π) σ)
    Z = (X − μ) / σ ~ 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 μ and σ 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 ≈ 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 ν = 4. Ask which one matches a market that has had several crashes in a century.
    3. Load the NVIDIA 7-sigma day and move ν 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 "μ ± 2σ" and check the empirical rule: 95.4% for the normal. With the t at ν = 4 the band holds about the same (95.3%), yet the chance of a move beyond 3σ 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.