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).

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

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

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

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 2,720 words, 18 min read) adds 3 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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