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}

Read more about this topic:  Inverse Transform Sampling

Famous quotes containing the words proof of, proof and/or correctness:

    “The fact that several men were able to become infatuated with that latrine is truly the proof of the decline of the men of this century.”
    —Charles Baudelaire (1821–1867)

    “A short letter to a distant friend is, in my opinion, an insult like that of a slight bow or cursory salutation—a proof of unwillingness to do much, even where there is a necessity of doing something.”
    —Samuel Johnson (1709–1784)

    “With impressive proof on all sides of magnificent progress, no one can rightly deny the fundamental correctness of our economic system.”
    —Herbert Hoover (1874–1964)