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:

    “Ah! I have penetrated to those meadows on the morning of many a first spring day, jumping from hummock to hummock, from willow root to willow root, when the wild river valley and the woods were bathed in so pure and bright a light as would have waked the dead, if they had been slumbering in their graves, as some suppose. There needs no stronger proof of immortality. All things must live in such a light. O Death, where was thy sting? O Grave, where was thy victory, then?”
    —Henry David Thoreau (1817–1862)

    “If some books are deemed most baneful and their sale forbid, how, then, with deadlier facts, not dreams of doting men? Those whom books will hurt will not be proof against events. Events, not books, should be forbid.”
    —Herman Melville (1819–1891)

    “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)