A uniform input X on [−1,1] is transformed into Y=X². Values near zero arise from both positive and negative inputs. A naive one-branch derivative calculation loses half the probability. Begin by computing probabilities from inverse images: Y≤1/4 exactly when X lies between −1/2 and 1/2, so this probability is 1/2.
Predict
Predict whether every measurable transformation of a density produces another density. A constant transformation gives a point mass, so the answer is no. The general object is the pushforward measure; a density is an additional representation that requires absolute continuity with respect to a chosen reference measure.
Experiment
For the squaring map, compare output thresholds and their preimages.
Output event
Input event
Probability
Y≤1/16
−1/4≤X≤1/4
1/4
Y≤1/4
−1/2≤X≤1/2
1/2
Y≤1
−1≤X≤1
1
Draw both branches of the inverse relation. Then calculate E(Y) directly as E(X²)=1/3 and check it using the output density. These are two representations of the same measure, not two unrelated recipes.
Understand
If T maps (X, A) measurably into (Y, B), define its pushforward measure by T#μ(B)=μ(T inverse (B)). Inverse images preserve disjoint unions, so countable additivity follows immediately from that of μ. For a nonnegative measurable h, the identity integral h d(T#μ)=integral h∘T dμ first holds for indicators by definition, then for simple functions by linearity, then for arbitrary nonnegative functions by increasing approximation and monotone convergence. For integrable signed functions apply it to the positive and negative parts. This is the abstract change-of-variable theorem, requiring no differentiability.
In the squaring example, the output distribution function is sqrt (y) for 0≤y≤1. Its derivative gives density 1/(2sqrt(y)) on (0,1); integrating that density over intervals reproduces the distribution function and therefore the measure. Each inverse branch contributes its input density times the absolute derivative of the inverse. The two contributions add. Endpoint values of a density do not matter, but endpoint atoms would matter and must be checked separately.
For a continuously differentiable strictly monotone map on an interval with nonzero derivative, the ordinary one-variable substitution theorem can first be proved for interval indicators from the fundamental theorem of calculus applied to the inverse. Uniqueness of the induced measures, followed by simple approximation, extends it to nonnegative measurable functions. A finite collection of monotone branches is handled by partitioning the input and summing these contributions; branch endpoints form a null set when the input has a density.
The higher-dimensional Jacobian substitution theorem requires an additional local geometric theorem about differentiable maps. This lesson proves the measure-theoretic transport identity and the stated one-dimensional branch mechanism; it does not derive that full geometric theorem from a symbolic determinant rule.
Apply
For a map that is constant on a positive-measure input region and monotone elsewhere, predict a mixed output law: an atom plus a density. Compute the atom from the mass of the constant region before transforming the remaining part. This prevents a common error in which a derivative formula silently discards concentrated mass.
Continue
Always distinguish a transformation, its pushforward measure and a density relative to a reference measure. Later Radon–Nikodym theory will characterize when such a density exists. The same inverse-image identity underlies distributions of random variables, sufficient statistics and changes of probability measure.
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Let X be uniform on [−1,1] and Y=X². What is P(Y≤1/9)?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For the same X and Y, compute E(Y²).
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Prove integral h d(T pushforward μ)=integral h∘T dμ for nonnegative measurable h.
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Derive the density of X² when X is uniform on [−1,1].
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Explain why a constant transformation cannot be described by an ordinary Lebesgue density.
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Prove (S∘T) pushforward μ equals S pushforward of T pushforward μ.
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Reflect and continue
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