Knowlpedia

Theorem

Probability integral transform

A continuous cumulative distribution function sends its random variable to the uniform law.

Statement

If a real random variable ZZ has continuous FF, then the probability integral transform states that F(Z)F(Z) is uniform on (0,1)(0,1).

Standard normal coordinates

For the strictly increasing standard normal CDF

Φ(t)=12π∫−∞te−s2/2 ds,\Phi(t)=\frac1{\sqrt{2\pi}}\int_{-\infty}^t e^{-s^2/2}\,ds,

the map Φ:R→(0,1)\Phi:\mathbb R\to(0,1) is a bijection, and P(Φ(Z)≤u)=P(Z≤Φ−1(u))=u\mathbb P(\Phi(Z)\leq u)=\mathbb P(Z\leq\Phi^{-1}(u))=u. Applying it coordinatewise gives a measure-space isomorphism

Fd:(Rd,γd)→((0,1)d,λd),Fd(x)j=Φ(xj).F_d:(\mathbb R^d,\gamma_d)\to((0,1)^d,\lambda_d), \qquad F_d(x)_j=\Phi(x_j).
Scope

A CDF with jumps does not generally produce a uniform variable by direct composition. Continuity is the hypothesis of the theorem. Strict monotonicity is the additional property used in the Gaussian example to obtain an ordinary inverse function.