Inverse Transform Sampling - Proof of Correctness

Proof of Correctness

Let F be a continuous cumulative distribution function, and let F−1 be its inverse function (using the infimum because CDFs are weakly monotonic and right-continuous):

Claim: If U is a uniform random variable on (0, 1) then follows the distribution F.

Proof:

$\begin{align} & \Pr(F^{-1}(U) \leq x) \\ & {} = \Pr(\inf\;\{y \mid F(y)=U\} \leq x)\quad &\text{(by definition of }F^{-1}) \\ & {} = \Pr(U \leq F(x)) \quad &\text{(applying }F,\text{ which is monotonic, to both sides)} \\ & {} = F(x)\quad &\text{(because }\Pr(U \leq y) = y,\text{ since }U\text{ is uniform on the unit interval)}. \end{align}$

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