Transformations, quantiles and inverse sampling

What this concept is

A transformation changes a random variable, not just its units. Begin with the CDF: F_Y(y)=P(g(X)≤y). For a differentiable strictly monotone transformation with differentiable inverse, the density uses the absolute inverse derivative wherever the formula is valid.

Why it matters here

This connects calculus to simulation and prevents incorrect percentile and density conversions. Finish CAL1–CAL6 before this cluster.

Concrete example

If X is uniform on [0,1] and Y=X², then F_Y(y)=sqrt(y) on [0,1], zero below and one above. Its density is 1/(2sqrt(y)) for 0<y<1. The density grows without bound near zero but its improper integral is 1. The mean is ∫₀¹ x² dx=1/3, not (E[X])²=1/4.

Common confusion / misconception

One-to-one formulas cannot be applied blindly to squaring across negative and positive inputs. A nonlinear transform of a mean is generally not the mean of the transform.

How to use it

For an exponential rate λ>0, F(t)=1−exp(−λt) on t≥0. Solve u=F(t): Q(u)=−ln(1−u)/λ for 0≤u<1. If U is ideal continuous uniform, P(Q(U)≤t)=F(t), which proves the model transformation. In code use −log1p(−u)/λ to avoid cancellation for tiny u; reject u=1 rather than silently returning a finite tail. Finite pseudorandom generators approximate this ideal and cannot produce every real tail value.

Check yourself

Why is Q(0)=0 but Q(1) not a finite sample? What is the median for rate 2?

Mini drill or application

For Y=3X+2 with X uniform on [0,1], derive support [2,5], density 1/3 and CDF (y−2)/3 on the support. For exponential rate 2, the median is ln(2)/2. Explain why a density needs a scale factor when changing units.

Read this only if stuck

Primary source support. See the extension guide for practice, scope and evidence requirements. Local examples and lab code are original.