Lévy is what happens when you weaken the CLT's finite-variance assumption — it generalises Normal to a family with stable α ∈ (0, 2]. Lévy's α = 1/2 is the extreme heavy-tailed special case.
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Worked example
The Lévy distribution (with location 0 and scale c = 1) is the special α-stable distribution with stability index α = 1/2 and full skewness β = 1. Its CDF is F(x) = erfc(sqrt(c / (2x))) for x > 0. (a) Compute P(X < 2). (b) Compute P(X > 100). (c) Show that E[X] is infinite. (d) State and verify the stability property under sums.
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Reflection
Where does this distribution sit in the story chain — what question does it answer that the previous distribution couldn't? Try to recall the key moment formulas (mean, variance) without looking them up.