The limit distribution of the maximum of many independent draws. Extreme-value statistics for floods, market crashes, record events.
Method · Gumbel
Intro
User:Herr blaschke โ public domain · Public Domain · Wikimedia Commons
Gumbel is the asymptotic distribution of the MAX of iid samples with exponential-like tails โ the natural successor to Normal once we ask 'what about the EXTREME, not the average?'
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Try first (productive failure)
Before the worked example: spend 60 seconds taking your best shot at this.
A guess is fine โ being briefly wrong about a problem makes the explanation
land harder when you read it. This appears once per tutorial; skip
if you already know the trick.
60s
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Worked example
Let X_1, ..., X_n be iid Exp(rate ฮป = 1). Let M_n = max(X_1, ..., X_n). (a) Derive the CDF of M_n. (b) Compute E[M_n] for n = 100. (c) Compare to the Gumbel asymptotic E[M_n] โ ln n + ฮณ_Euler, where ฮณ_Euler โ 0.5772. (d) Compute P(M_50 > 5).
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Practice 1 of 3Type a fraction, decimal, or expression.
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Reflection
Where does this distribution sit in the story chain โ what question does it answer that the previous distribution couldn't? Try to recall the key moment formulas (mean, variance) without looking them up.
