Mixed Discrete and Continuous Loss Distribution¶

Definition¶

A mixed discrete and continuous loss has both positive point masses and an absolutely continuous component. Its distribution can be written $\displaystyle \mathbb{P}(X\in A)=\sum_jp_j\indicator_{\{a_j\in A\}}+\int_A f_c(x)\,\mathrm{d}x$${\int }_A{f}_c(x)\mathrm{d}x$, where $p_j>0$ and $\displaystyle \sum_jp_j+\int f_c=1$$1$.

Interpretation¶

A claim can be zero, a normal positive amount, or exactly a policy cap. Jumps of the cumulative distribution function (CDF) record point masses; the ordinary derivative between jumps records only the continuous component. This is a specific distribution structure, while a mixture describes averaging component laws.

The figure shows the example below. The CDF jumps from 0 to $0.8$ at $x=0$$0$, and the jump height is the point mass $\mathbb{P}(X=0)=F_X(0)-F_X(0-)=0.8$$0.8$. Between 0 and 100 the CDF rises linearly from $0.8$ to 1 with slope $0.2/100=0.002$$0.002$, the continuous component density $f_c$. No single ordinary density can produce the jump, so both pieces are needed.

A mixture distribution can combine a discrete component and an absolutely continuous component. The component weights give the total atom mass and the total continuous mass; the example mixes a point mass at zero with a uniform positive loss.

Key Results¶

• $\mathbb{P}(X=a)=F_X(a)-F_X(a-)$$F_X(a-)$.
• $\displaystyle \mathbb{E}[h(X)]=\sum_jp_jh(a_j)+\int h(x)f_c(x)\,\mathrm{d}x$ when integrable.
• A single ordinary density with respect to length cannot represent the atoms.
• Coverage modifications can introduce atoms even when the original loss is continuous.

Examples¶

If $X$ is zero with probability $0.8$ and otherwise uniform on $(0,100)$, its continuous component density is $0.002$ on that interval. The mean is $0.2(50)=10$$10$.

Prerequisites¶

• Point Mass

• Absolutely Continuous Distribution

• Mixture Distribution

• Cumulative Distribution Function

Related Concepts¶

• Ordinary Deductible
• Policy Limit
• Franchise Deductible
• Individual Risk Model

Sources¶

• Risk Insurance · Lecture 1

Aliases: mixed loss distribution, loss distribution with atoms, mixed distribution, mixed random variable Keywords: point mass at zero, atom at the policy limit, jump of the CDF, P(X = a) = F(a) - F(a-), continuous component density Symbols: element of, integral, sum

Last updated: 16 October 2026 22:28