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Pareto distribution

A heavy-tailed positive loss where large claims stay meaningfully likely.

Parameters and support

$\alpha$

shape — larger means a lighter tail — $\alpha >0$

$\theta$

scale — $\theta >0$

Support

$x>0$

The formulas

f(x)

$$f(x)=\frac{\alpha {\theta }^{\alpha }}{(x+\theta {)}^{\alpha +1}}$$

F(x)

$$F(x)=1-{\left(\frac{\theta }{x+\theta }\right)}^{\alpha }$$

Mean

$$E[X]=\frac{\theta }{\alpha -1},\alpha >1$$

Variance

$$\mathrm{Var}(X)=\frac{\alpha {\theta }^2}{(\alpha -1{)}^2(\alpha -2)},\alpha >2$$

MGF

Not examinable for this distribution.

No MGF exists — the tail is heavier than any exponential.

Where the moments come from

1. $S(x)={\left(\frac{\theta }{x+\theta }\right)}^{\alpha }$ decays polynomially, not exponentially.
2. $E[X]={\int }_0^{\infty }S(x)dx={\theta }^{\alpha }{\int }_0^{\infty }(x+\theta {)}^{-\alpha }dx$, which converges only when $\alpha >1$, giving $\frac{\theta }{\alpha -1}$.
3. $E[X^2]={\int }_0^{\infty }2xS(x)dx=\frac{2{\theta }^2}{(\alpha -1)(\alpha -2)}$ for $\alpha >2$.
4. Subtracting the squared mean gives $\frac{\alpha {\theta }^2}{(\alpha -1{)}^2(\alpha -2)}$.

Worked example

The mean, variance, CDF and moment generating function above are re-derived numerically from this distribution’s own density on every test run — summed over the support for a discrete distribution, integrated by quadrature for a continuous one — and compared with the closed forms printed here. A typo on this page fails the build.

Traps

• Quoting a mean when α ≤ 1 or a variance when α ≤ 2 — they are infinite, and 'does not exist' is the answer.
• Confusing this two-parameter form with the single-parameter Pareto whose support starts at θ.

Related

• Exponential distribution
• Lognormal distribution

Drill this: the Exam P question bank has original questions on this distribution, and the recall trainer builds its prompts from exactly the formulas above.