Functions of Random Variables

The change-of-variables machinery for transformed random variables is used constantly across mathematics and its applications. In statistics, it derives the sampling distributions of test statistics: the square of a standard normal is $\chi^2_1$ (a direct instance of the piecewise-monotonic corollary above, specialized to a symmetric $X$), sums of such squares give $\chi^2_n$, and ratios of independent chi-squared variables give the $F$-distribution used in ANOVA. In simulation and computer graphics, the inverse CDF method (foreshadowed here) reverses the roles: given $U\sim \mathrm{Uniform}(0,1)$, the variable $X = F_X^{-1}(U)$ has CDF exactly $F_X$, because $F_X^{-1}$ is itself the transformation $g$ in the theorem, run through the monotonic-preimage lemma — this is precisely how software libraries generate exponential, normal, and other non-uniform random variates from a uniform random number generator. In physics, transformations of random variables appear whenever a measured quantity (energy, momentum, radius) is a nonlinear function of a fundamentally random quantity, such as converting a uniformly distributed scattering angle into a non-uniformly distributed detector position — the Jacobian factor is literally the "phase space" volume element familiar from statistical mechanics. In finance, the log-normal model for asset prices is obtained by applying $g(x)=e^x$ to a normal random variable $X$, an instance of the strictly-monotonic change-of-variables theorem with $g$ increasing and $g^{-1}(y)=\ln y$. In cryptography and coding theory, transformations of random bit-strings (hash outputs, error patterns) are analyzed via the discrete analogue of the pushforward construction. In machine learning, normalizing flows — a modern generative modeling technique — are built entirely on iterated applications of the multivariate change-of-variables formula, using compositions of invertible neural-network layers whose Jacobian determinants must be tracked exactly as in this article's single-variable case.

Ask the tutor: How does the single-variable change-of-variables formula generalize to a joint transformation of two or more random variables using the Jacobian determinant?

Think of $X$ as a machine that outputs a real number according to some law, and think of $g$ as a lens through which you view that output. You never see $X$ directly — you only see $Y = g(X)$, the reading after passing through the lens. The distribution of $Y$ answers: "given the probabilities governing $X$, what probabilities govern what I actually observe?" The CDF method is the event-level answer: to know $P(Y \le y)$, find the set of $X$-values that the lens maps into $(-\infty, y]$, and compute the $X$-probability of that set. This works for any measurable $g$, including non-monotonic, non-differentiable, or even discontinuous lenses — a step function, an absolute value, a clipping function. The change-of-variables method is the density-level refinement, valid when $g$ is smooth and locally invertible. Density is probability mass per unit length. If $g$ stretches a small interval around $x$ by a factor of $|g'(x)|$, then the same amount of probability mass gets smeared over a longer (or shorter) interval around $y = g(x)$, so the density must be divided by that stretch factor to conserve total mass. This is exactly why the Jacobian $|(g^{-1})'(y)|$ appears: it measures how much the inverse map compresses or expands the $y$-axis back into $x$-space.

This theorem is completely general — it places no hypothesis on $g$ beyond measurability, which is needed merely for $Y$ to be a legitimate random variable. Its cost is that $A_y$ can be an arbitrarily complicated set, and computing $\int_{A_y} f_X$ may require case analysis. The next theorem specializes to the situation where $g$ is smooth and monotonic, in which case $A_y$ is simply a ray, and differentiating $F_Y$ produces a clean formula for $f_Y$.

Example 1 (asymmetric linear-then-nonlinear transformation, using the CDF method to sanity-check the Jacobian method). Let $X \sim \mathrm{Exponential}(\lambda)$ with $\lambda = 2$, so $f_X(x) = 2e^{-2x}$ for $x>0$ and $F_X(x) = 1-e^{-2x}$ for $x > 0$. Let

an asymmetric affine transformation (not just a rescaling): it shifts by $3$, flips sign, and stretches by $5$. Find $f_Y$. CDF method. Since $X>0$ a.s., $Y = 3-5X < 3$ a.s. For $y < 3$,

For $y \ge 3$, $F_Y(y) = 1$ (in fact this formula extended continuously gives $1$ at $y=3$, consistent). Differentiating for $y<3$:

and $f_Y(y)=0$ for $y \ge 3$. Jacobian method, as a check. Here $g(x) = 3-5x$ is strictly decreasing on $I=(0,\infty)$, with inverse $g^{-1}(y) = \frac{3-y}{5}$ and $(g^{-1})'(y) = -\frac15$, so $|(g^{-1})'(y)| = \frac15$. The range is $J = g((0,\infty)) = (-\infty,3)$. By the theorem,

matching the CDF-method answer exactly. As a numerical check, at $y=-2$: $\frac{3-y}{5} = 1$, so $f_Y(-2) = \frac25 e^{-2} \approx 0.4\times 0.1353 = 0.05413$; total mass check $\int_{-\infty}^3 f_Y(y)\,dy = \int_0^\infty f_X(x)\,dx = 1$ by substitution, as it must.

Example 2 (non-monotonic transformation, needing the piecewise-monotonic corollary). Let $X \sim N(1,4)$ (mean $1$, variance $4$, so $\sigma=2$), with density

Let $Y = X^2$. Since $X$ is not centered at $0$, this is a genuinely asymmetric instance of squaring — the two branches contribute unequally. Find $f_Y(y)$ for $y > 0$. Here $g(x) = x^2$ is not globally monotonic on $\mathbb R$, but splits into two strictly monotonic branches: $I_1 = (-\infty, 0)$ with $g_1(x) = x^2$ strictly decreasing, inverse $g_1^{-1}(y) = -\sqrt y$; and $I_2 = (0,\infty)$ with $g_2(x)=x^2$ strictly increasing, inverse $g_2^{-1}(y) = \sqrt y$. Both map onto $J = (0,\infty)$. In each case $|(g_k^{-1})'(y)| = \left|\pm\frac{1}{2\sqrt y}\right| = \frac{1}{2\sqrt y}$. By the Piecewise-monotonic change of variables corollary, for $y>0$:

Plugging in $f_X$:

Note the two exponential terms are not equal (unlike the standard-normal case $X\sim N(0,1)$, where they'd coincide and give the familiar $\chi^2_1$ density) — this asymmetry is exactly the point of choosing mean $1\neq 0$. As a numerical spot check at $y=1$ (so $\sqrt y = 1$): the two exponents are $-\frac{(1-1)^2}{8}=0$ and $-\frac{(-1-1)^2}{8} = -\frac{4}{8}=-0.5$, giving

One can confirm this is sensible: since $f_X(\sqrt y)$ dominates $f_X(-\sqrt y)$ whenever $\sqrt y$ is close to the mean $1$ (as here), the "positive branch" contributes the bulk of the density, illustrating exactly why the naive shortcut of doubling one branch (valid only when $X$ is symmetric about $0$) would be wrong here.

The change-of-variables technique for densities is a direct descendant of the classical change-of-variables formula in multivariable calculus (the Jacobian determinant rule), transplanted into probability theory as measure-theoretic probability matured in the early twentieth century. No single person is credited with "inventing" the technique for random variables — it is a natural corollary of the measure-theoretic framework for probability — but its systematic, rigorous formulation is inseparable from the axiomatization of probability theory due to Andrei Kolmogorov, whose 1933 monograph Grundbegriffe der Wahrscheinlichkeitsrechnung placed random variables on the same footing as measurable functions in analysis, making constructions like pushforward measures (and hence the CDF method) precise objects rather than heuristic manipulations.