\[ \newcommand{\Exg}{\operatorname{\mathbb{E}}} \newcommand{\Ex}{\mathbb{E}} \newcommand{\Ind}{\mathbb{I}} \newcommand{\Var}{\operatorname{Var}} \newcommand{\Cov}{\operatorname{Cov}} \newcommand{\Corr}{\operatorname{Corr}} \newcommand{\ee}{\mathrm{e}} \]

13  Inverse transform method

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Summary:

• The inverse \(F^{-1}\) of a CDF \(F\) is defined by \(F^{-1}_X(u) = \min \big \{x : F_X(x) \geq u \big\}\).

• The inverse transform method converts \(U \sim \operatorname{U}[0,1]\) to a random variable with CDF by setting \(X = F^{-1}(U)\). That is: Set \(U = F(X)\), and rearrange to make \(X\) the subject.

• The Box–Muller transform is a way to generate two independent standard normal distributions. Set \(R\) to Rayleigh with scale parameter 1, set \(\Theta \sim \operatorname{U}[0,2\pi]\), then take \(X = R \cos \Theta\) and \(Y = R \sin \Theta\).

Read more: Voss, An Introduction to Statistical Computing, Section 1.3.