Conditional CDFs and Interval Probabilities

From a Joint Density to a Conditional CDF

A joint density describes the probability of ranges of values for two random variables together. To find the conditional distribution of \(X\) at a given value \(y\) of \(Y\), focus on the horizontal slice of that joint density at \(Y=y\). Integrating along this slice up to \(x\) accumulates the conditional probability that \(X\) is no greater than \(x\). The slice must be normalized by the marginal density of \(Y\) at \(y\).

This construction is the CDF counterpart of the conditional-density formula: integrate the conditional density from the lower end of the real line to \(x\). The resulting function is useful both for describing the full conditional distribution and for calculating probabilities over intervals.

The qualifier “almost every” matters. A density is defined only up to changes on sets of Lebesgue measure zero, and conditioning on a particular value of a continuous random variable does not, in general, determine a unique conditional law at every single \(y\). The slice formula gives a version of the conditional distribution for almost every \(y\) where the marginal density is positive and finite. In examples with a natural, explicitly defined density, the formula will usually be used throughout the indicated range of \(y\).

For any fixed \(y\) where the formula applies, its numerator integrates the joint density only over points with second coordinate \(y\), from \(u=-\infty\) through \(u=x\). Dividing by \(f_Y(y)\) turns that accumulated slice into a conditional probability. In particular, integrating the whole slice gives \(f_Y(y)\), so the conditional CDF tends to 1 as \(x\) increases without bound.

Conditional Interval Probabilities

A CDF gives interval probabilities by taking differences. For real numbers \(a<b\),

The second equality follows by subtracting the two integrals defining the CDF; the part from \(-\infty\) to \(a\) cancels. Because the conditional law obtained from a density has no point masses for almost every such \(y\), changing either interval endpoint from open to closed does not change its probability for those \(y\). Stating the interval as \((a,b]\) is convenient because it matches the standard CDF difference exactly.

Proof. Since the joint density is nonnegative and integrates to 1, Tonelli’s theorem gives \(\int_{\mathbb{R}} f_Y(y)\,dy=1\). Thus \(f_Y(y)\) is finite for almost every \(y\). For a \(y\) with \(0<f_Y(y)<\infty\), set \(q_y(C)=\int_C f_{X,Y}(u,y)\,du/f_Y(y)\) for each Borel set \(C\subseteq\mathbb{R}\), whenever the slice integral is defined. The integral over the entire real line is \(f_Y(y)\), so \(q_y(\mathbb{R})=1\); nonnegativity and countable additivity follow from those properties of the integral. Hence \(q_y\) is a probability distribution.

To check that this distribution is conditional, let \(A\) be a Borel set of possible values for \(Y\), and let \(C\) be a Borel set of possible values for \(X\). Fubini’s theorem applies to the nonnegative joint density and gives

This identity is the defining conditional-distribution identity. It holds for all Borel \(C\) and \(A\) after choosing the conditional distribution version outside a single null set of \(y\)'s; this follows first for a countable generating class of intervals and then for Borel sets by the standard uniqueness and extension argument for probability measures. Taking \(C=(-\infty,x]\) gives the stated CDF formula. Taking \(C=(a,b]\) gives the interval formula. \(\square\)

Worked Calculations

Recovering the Conditional Density

The slice formula also explains how a conditional density can be recovered from its CDF. For a fixed admissible \(y\), the conditional CDF is an integral of the slice density divided by the marginal density. The fundamental theorem of calculus for Lebesgue integrals says that the derivative of an indefinite integral of an integrable function equals that function almost everywhere. Consequently, the conditional density is the normalized slice density almost everywhere in \(x\).

Proof. For any \(r<s\), the CDF formula gives

The integrand is nonnegative and integrable over \(\mathbb{R}\), with integral 1. An indefinite integral of an integrable function is absolutely continuous, and its derivative equals its integrand at almost every point. Applying this fundamental theorem of calculus yields the asserted conditional density. \(\square\)

Common Mistake: Forgetting to Normalize the Slice

A frequent error is to integrate the joint density along a slice and call the result a conditional CDF without dividing by the marginal density. In the triangular example, the unnormalized accumulation for \(y<x<1\) is \(2(x-y)\). At \(x=1\), this equals \(2(1-y)\), not 1; it is the marginal density \(f_Y(y)\), not a conditional probability. Dividing by \(2(1-y)\) produces \((x-y)/(1-y)\), which reaches 1 at the right endpoint, as a CDF must.

A second check is to distinguish an interval probability from a density value. The CDF difference \(F_{X\mid Y}(b\mid y)-F_{X\mid Y}(a\mid y)\) is a probability. A conditional density is instead the derivative of the CDF where that derivative exists; it is not itself the probability of an interval.