Distributions as pushforward measures

Given a measurable map X, define its law by assigning a target event B the probability that X lands in B. The relevant original event is the inverse image of B. Since inverse images preserve complements and countable disjoint unions, the probability axioms transfer to the new measure. No inverse function for X is needed. The map may identify many different outcomes, and the resulting distribution simply combines their probabilities when they produce the same observable value.

For a real random variable, the cumulative distribution function records the probability of each lower half-line. It is nondecreasing, right-continuous, and tends to zero and one at the two infinities. These values determine the Borel probability measure uniquely. A jump represents a point mass. A continuous cumulative distribution function need not possess a density, so differentiating it and assuming the derivative recovers all its probability is not justified without an absolute-continuity hypothesis.

If Y=g(X) for a measurable function g, its law is the pushforward of the law of X through g. To compute it, describe the set of x values whose transformed value lies in the requested target set. This method works when g is noninjective, where using a single inverse branch loses probability. For a squaring map both positive and negative inputs may contribute to one positive output. Atoms also survive or merge according to the same inverse-image rule.

Integration follows the same transport principle: an expectation of a nonnegative measurable function of X can be integrated against its law. For signed functions the identity applies when the absolute value is integrable, or in the appropriate extended sense when positive and negative parts do not both have infinite integrals. This result is often called the transfer formula. It is more general than a density-based substitution and gives a reliable way to compute expectations for mixtures or transformed discrete variables.

Take a fair sign variable X with values −1 and 1. Setting Y=X and setting Z=−X gives Y and Z exactly the same marginal distribution. Nevertheless X+Y is either −2 or 2, whereas X+Z is always zero. The difference lives in the joint distribution on the product space. A marginal law describes what one coordinate does alone; a coupling specifies how coordinates occur together. Independent coupling is one choice, not an automatic consequence of naming two distributions.

More generally, two variables defined on different probability spaces can have the same law even when comparing their samplewise values makes no sense. Equality in distribution is therefore weaker than almost-sure equality of variables on one common space. When a calculation involves sums, maxima, or simultaneous events, first check whether the joint law is available. Marginal distributions determine expectations of separate integrable terms by linearity, but usually do not determine probabilities of joint events or variances of sums.