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:

    If we view our children as stupid, naughty, disturbed, or guilty of their misdeeds, they will learn to behold themselves as foolish, faulty, or shameful specimens of humanity. They will regard us as judges from whom they wish to hide, and they will interpret everything we say as further proof of their unworthiness. If we view them as innocent, or at least merely ignorant, they will gain understanding from their experiences, and they will continue to regard us as wise partners.
    Polly Berrien Berends (20th century)

    The source of Pyrrhonism comes from failing to distinguish between a demonstration, a proof and a probability. A demonstration supposes that the contradictory idea is impossible; a proof of fact is where all the reasons lead to belief, without there being any pretext for doubt; a probability is where the reasons for belief are stronger than those for doubting.
    Andrew Michael Ramsay (1686–1743)

    The surest guide to the correctness of the path that women take is joy in the struggle. Revolution is the festival of the oppressed.
    Germaine Greer (b. 1939)