6  How Fast Does the Tail Die?

Introduces tail decay as a spectrum rather than a binary:

def gaussian_tail(x):
    from math import exp
    return exp(-x**2 / 2)          # decays super-fast

def exponential_tail(x, lam=1):
    from math import exp
    return exp(-lam * x)           # decays fast

def pareto_tail(x, x_min=1, alpha=2):
    return (x_min / x) ** alpha    # decays slowly (polynomially)

Plotting all three survival functions on the same log-log axes makes the qualitative difference vivid: Gaussian and exponential curves fall off a cliff; the Pareto curve just keeps sloping down. The organizing question for the rest of the book — “how quickly does \(\bar F(x)\) go to zero, and what does that speed cost you?” — is introduced here explicitly.