Inverse transform sampling

Creator
Creator
Seonglae Cho
Created
Created
2024 Nov 27 16:33
Editor
Edited
Edited
2025 Apr 29 0:39
Refs
Refs

Inversion sampling, inverse probability integral transform

Need to know the CDF analytically and have a tractable inverse FX1F_X^{-1}.
  1. Compute
    Cumulative Distribution Function
    FXF_X
  1. Generate random samples from
    Uniform distribution
    UU(0,1)U \sim \mathcal{U}(0, 1)
  1. Inverse transform using x=F1(u)x=F^{-1} (u) to sample xx then V=FX1(U)=XV = F_X^{-1} (U) = X
The distribution of the random variable VV is the same as XX.

Proof

First of all, FX(X)U(0,1)F_X(X) \sim \mathcal{U}(0,1)
CDF(V)=P(FX1(U)<t)=P(U<FX(t))=FX(t)fU(z)dz=FX(t). \text{CDF}(V) = P(F_X^{-1}(U) < t) = P(U < F_X(t)) = \int_{-\infty}^{F_X(t)} f_U(z) \, dz = F_X(t). 
 
 
 
 

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