Simulate random variables using the inversion method and basic Monte Carlo techniques.

A reinsurer needs the 99.5% VaR of next year's aggregate losses, and the distribution has no closed form. You write a loop that draws uniform numbers, converts each one into a claim, sums portfolios, and reads off an empirical quantile. That loop is Monte Carlo, and the conversion step is inversion.

Why inversion works. Define the generalized inverse $F^{-1}(u)=\inf\{x:F(x)\ge u\}$ (needed for step CDFs). If $U\sim\text{Uniform}(0,1)$, then $P(F^{-1}(U)\le x)=P(U\le F(x))=F(x)$. So $X=F^{-1}(U)$ has the target distribution.

KEY: One uniform draw produces exactly one target draw. Inversion is the gold-standard method when $F^{-1}$ is available in closed form or by quick numeric search.

$$X = -\frac{1}{\lambda}\ln(1-U) \quad\text{for}\quad X\sim\text{Exponential}(\lambda)$$

Because $1-U$ is also uniform, programmers often write $X=-\ln(U)/\lambda$; both are valid. For Weibull$(\alpha,\theta)$: $X=\theta(-\ln(1-U))^{1/\alpha}$. For Pareto$(\alpha,\theta)$: $X=\theta((1-U)^{-1/\alpha}-1)$.

Discrete inversion. Order the support $x_1<x_2<\cdots$ and form cumulative probabilities. Draw $U$, then return the smallest $x_k$ with $F(x_k)\ge U$.

EXAMPLE: A claim count is 0 w.p. 0.5, 1 w.p. 0.3, 2 w.p. 0.2. Cumulative: 0.5, 0.8, 1.0. If $U=0.62$, return 1. If $U=0.81$, return 2.

Common mistakes

• Using $F(U)$ instead of $F^{-1}(U)$. The probability integral transform says that if $X$ has continuous CDF $F$, then $F(X)$ is uniform on $(0,1)$. Applying $F$ to a uniform $U$ generally produces neither a uniform variable nor a draw from $F$. To generate a draw from $F$, use $F^{-1}(U)$, with the generalized inverse for discrete distributions.
• Forgetting the generalized inverse for discrete $F$. Take the smallest $x_k$ with $F(x_k)\ge U$. Strict inequality or interpolation produces wrong support points.
• Reporting $s/n$ as the standard error. Monte Carlo SE is $s/\sqrt{n}$. Dividing by $n$ understates uncertainty by $\sqrt{n}$.

Bottom line

• Inversion method: if $U\sim\text{Uniform}(0,1)$, then $X=F^{-1}(U)$ has CDF $F$; works for any $F$, continuous or discrete, using the generalized inverse (smallest $x_k$ with $F(x_k)\ge U$) at jumps.
• Exponential inversion $X=-\ln(1-U)/\lambda$ is the building block for Weibull and Pareto draws.
• Monte Carlo mean estimator: $\hat\theta=\frac{1}{n}\sum g(X_i)$; unbiased, with standard error $s/\sqrt{n}$.
• Convergence rate is $1/\sqrt{n}$: to halve the error you quadruple the sample.

Exam shortcut

When a question hands you a CDF and a uniform, write $X=F^{-1}(U)$ symbolically and solve before computing; exponential, Pareto, Weibull, and uniform on $[a,b]$ all have one-line inverses. To size a Monte Carlo run, use $n=(1.96\,s/h)^2$ with the pilot SD whenever a target half-width is given.

The full lesson (about 1,549 words, 10 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• A7

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