$X \sim F_X(x), \quad x \geq 0$ The Loss Random Variable ($X$) represents the total ground-up loss amount arising from an insured event, before any policy modifications such as deductibles, coinsurance, or benefit limits are applied. $X$ is always non-negative and its distribution (commonly exponential, lognormal, gamma, or Pareto) models the underlying severity of losses. It serves as the input to the payment function that determines what the insurer actually pays. > [!example]- Moments of the Loss Random Variable {💡 Example} > Ground-up losses follow $X \sim \text{Exponential}(\theta = 2000)$. Find $E[X]$, $E[X^2]$, and $\text{Var}(X)$. > > > [!answer]- Answer > > For an exponential distribution with mean $\theta$: $E[X] = \theta$ and $E[X^2] = 2\theta^2$. > > $E[X] = 2000$ > > $E[X^2] = 2(2000)^2 = 8{,}000{,}000$ > > $\text{Var}(X) = E[X^2] - (E[X])^2 = 8{,}000{,}000 - 4{,}000{,}000 = 4{,}000{,}000$