Common Parametric Distributions

The exam gives you "Losses follow a Pareto with $\alpha =3$ and $\theta =1000$." If you do not know the mean formula instantly ($\theta /(\alpha -1)=500$) you waste two minutes deriving it. The distributions on this list appear in 60% or more of Exam P problems.

$X\sim \text{Exp}(\theta )$ with scale parameter $\theta >0$ (mean parametrization).

$$f(x)=\frac{1}{\theta }{e}^{-x/\theta },x>0$$

$$F(x)=1-e^{-x/\theta },S(x)=e^{-x/\theta }$$

$E[X]=\theta$, $\text{Var}(X)={\theta }^2$, $\text{CV}(X)=1$.

HIGH-FREQUENCY: The memoryless property: $P(X>s+t\mid X>s)=P(X>t)$. The exponential is the only continuous distribution with this property. Given survival past $s$, the remaining lifetime has the same distribution as the original.

Key LEV: $E[X\wedge d]=\theta (1-e^{-d/\theta })$.

Common mistakes

• Confusing Pareto parameters when the mean does not exist. For $\alpha =1$, $E[X]=\theta /(\alpha -1)$ is undefined (division by zero). The LEV still exists for any $d>0$, but the mean and per-loss expected payment do not.
• Using the wrong Pareto survival function. The SOA Pareto has $S(x)=[\theta /(x+\theta ){]}^{\alpha }$. Some textbooks use $S(x)=(\theta /x{)}^{\alpha }$ for $x>\theta$ (the "single-parameter" Pareto). Mixing formulas produces wrong probabilities.
• Forgetting $\alpha >2$ is needed for the Pareto variance. Computing $\text{Var}(X)$ with $\alpha =1.5$ gives a formula that evaluates to a negative denominator. The variance does not exist.

Bottom line

• Exponential: memoryless with CV = 1, mean $\theta$, variance ${\theta }^2$, CDF $1-e^{-x/\theta }$
• Uniform: constant density, mean $(a+b)/2$, variance $(b-a{)}^2/12$
• Normal: symmetric bell, fully specified by $\mu$ and ${\sigma }^2$, closed under linear combinations
• Gamma: sum of $\alpha$ iid exponentials (same scale), mean $\alpha \theta$

Exam shortcut

Memorize four LEV formulas: Exponential $\theta (1-e^{-d/\theta })$, Uniform $d-d^2/(2b)$, Pareto $\frac{\theta }{\alpha -1}[1-(\theta /(d+\theta ){)}^{\alpha -1}]$, and capped-at-support distributions (LEV = mean when $d$ exceeds the upper bound). "Gamma = sum of exponentials (same scale)." Use this to quickly convert between the two families. "Pareto mean needs alpha > 1, variance needs alpha > 2." Check the parameter constraint before computing moments.

The full lesson (about 2,319 words, 15 min read) adds 2 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 2a
• 2b
• 2c
• 2d
• 2e
• 2f

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