Pushforward Laws and One-to-One Transformations

Transforming a Random Variable Means Transforming Its Law

A transformation \(Y=g(X)\) changes the values taken by a random variable, but it does not create a new source of randomness. The probabilities for \(Y\) come from the probabilities already assigned to \(X\). The precise link is made by asking which values of \(X\) would produce an outcome of \(Y\) in a specified event.

This viewpoint works whether \(X\) is discrete, continuous, or has a more complicated distribution. In particular, it does not require a probability density. The central operation is to take the inverse image of an event under \(g\).

Measurability matters: it ensures that \(g^{-1}(B)\) is an event whenever \(B\) is Borel, so the probability in the definition exists. The law of \(Y\) is entirely specified by the probabilities of these inverse images. If the original law is known, probabilities for transformed events can therefore be calculated by tracing the event back to \(X\).

What One-to-One Transformations Add

Every measurable transformation gives the inverse-image probability identity above. A one-to-one transformation adds the possibility of recovering which input values produced an output. That recovery must be handled with care: an injective map has an inverse function on its image, but event translation also requires the relevant image sets to be measurable.

To prove the event identity on \(\{X\in S\}\), first suppose \(X\in A\). Since \(Y=g(X)\), it follows that \(Y\in g(A)\). Conversely, if \(Y\in g(A)\), then \(Y=g(a)\) for some \(a\in A\). Also \(Y=g(X)\), with \(X\in S\). Injectivity of \(g\) on \(S\) gives \(X=a\), so \(X\in A\). Thus the two events agree whenever \(X\in S\). The excluded event has probability zero by assumption, so their probabilities agree. The image \(g(A)\) is Borel because it is the inverse image of \(A\) under the measurable inverse \(g^{-1}:T\to S\). This proves the result. \(\square\)

The bijection and measurable-inverse conditions are a useful sufficient set of assumptions. In particular, they hold for continuous strictly monotone functions on an interval, with their ranges as the sets \(T\). If one only needs the probability of an output event \(B\), no inverse function is needed: use \(P(Y\in B)=P(X\in g^{-1}(B))\) directly.

The Pushforward Law Is a Probability Law

The inverse-image definition does more than give a convenient calculation rule. It guarantees that the transformed probabilities satisfy the axioms of a probability distribution.

For the empty set, \(g^{-1}(\varnothing)=\varnothing\), so \(\mu_Y(\varnothing)=0\). Also \(g^{-1}(\mathbb R)=\mathbb R\), giving \(\mu_Y(\mathbb R)=\mu_X(\mathbb R)=1\). Now let \(B_1,B_2,\ldots\) be pairwise disjoint Borel sets. Their inverse images are pairwise disjoint: an \(x\) belonging to both \(g^{-1}(B_i)\) and \(g^{-1}(B_j)\) would have \(g(x)\in B_i\cap B_j\), which is empty when \(i\ne j\). Moreover, $$ g^{-1}\left(\bigcup_{i=1}^{\infty}B_i\right) = \bigcup_{i=1}^{\infty}g^{-1}(B_i). $$

Countable additivity of \(\mu_X\) therefore gives $$ \mu_Y\left(\bigcup_{i=1}^{\infty}B_i\right) = \mu_X\left(\bigcup_{i=1}^{\infty}g^{-1}(B_i)\right) = \sum_{i=1}^{\infty}\mu_X\bigl(g^{-1}(B_i)\bigr) = \sum_{i=1}^{\infty}\mu_Y(B_i). $$ Thus \(\mu_Y\) is nonnegative, has total mass one, and is countably additive, as required. \(\square\)

Expectations Can Also Be Transferred

The same idea extends from event probabilities to expectations. Once the law of \(Y\) is defined as a pushforward, integrating a function of \(Y\) is equivalent to integrating the composed function of \(X\). This identity can be useful even when the transformed law has no convenient density.

For an indicator function \(h(y)=\mathbf{1}_B(y)\), the identity is exactly the pushforward definition: $$ E[\mathbf{1}_B(Y)] = P(Y\in B) = P(X\in g^{-1}(B)) = E[\mathbf{1}_{g^{-1}(B)}(X)] = E[\mathbf{1}_B(g(X))]. $$

By finite linearity, the identity follows for every nonnegative simple function \(h=\sum_{j=1}^m a_j\mathbf{1}_{B_j}\), where \(a_j\geq 0\). For general nonnegative measurable \(h\), there is an increasing sequence of nonnegative simple functions \(h_k\) converging pointwise to \(h\). The identity holds for each \(h_k\). Applying the monotone convergence theorem to both sides gives $$ E[h(Y)] = \lim_{k\to\infty}E[h_k(Y)] = \lim_{k\to\infty}E[h_k(g(X))] = E[h(g(X))]. $$ For a signed integrable \(h\), apply the nonnegative result separately to \(h^+=\max(h,0)\) and \(h^-=\max(-h,0)\), then subtract their finite expectations. This proves the identity under the stated conditions. \(\square\)

When One-to-One Reasoning Fails

A common mistake is to assume that every transformation can be reversed. For a general map, the safe formula remains \(P(Y\in B)=P(X\in g^{-1}(B))\). But an event about \(X\) may not be recoverable from \(Y\): distinct input values can produce the same output.

The distinction is useful in practice. Use inverse images to calculate probabilities of events involving \(Y\), even for non-injective maps. Use a measurable inverse when the aim is to translate events about \(X\) into events about \(Y\). Neither approach requires assuming that \(X\) or \(Y\) has a density.