Identifying Components from a Hybrid CDF

Reading the Components of a Hybrid CDF

A cumulative distribution function can encode different kinds of probability in one formula. A jump represents probability concentrated at a single value, while a smooth increase can represent probability spread continuously over an interval. To identify the components correctly, read the CDF from left to right: compare its left and right values at candidate jump locations, then examine how it changes between those locations.

The key is to keep the two calculations distinct. At a point, the size of a jump gives the atom’s probability. On an interval where the CDF is absolutely continuous, its derivative gives the density (almost everywhere). Finally, add the atom probabilities and integrate the density to check that the total probability is one.

Right-continuity is part of the definition of a CDF, so \(F(x)\) gives the value after any jump at \(x\). The value just before that jump is \(F(x-)\). This distinction matters at boundaries as well as at interior points: if a CDF jumps at the left endpoint of a density interval, the jump contributes an atom, while the increase along the interval contributes continuous probability.

A Systematic Recovery Method

Suppose a CDF is specified by finitely many pieces. First list every point where the formula jumps, and compute \(F(x)-F(x-)\) at each one. Next, differentiate on each smooth piece that is absolutely continuous. The resulting density is zero outside the regions where the CDF increases continuously. Then verify that the atom sizes and integrated density account for one unit of probability.

Why the Probability Check Works

The total-probability check is more than a convenient arithmetic test. Under the appropriate regularity conditions, it follows directly by adding the continuous increases and the jumps. Absolute continuity on the pieces ensures that each continuous increase equals the integral of the derivative there.

Proof. Nondecreasingness of a CDF implies that each jump \(F(x_i)-F(x_i-)\) is nonnegative. On each interval without a jump, absolute continuity and the fundamental theorem of calculus identify the integral of \(f\) with the increase in \(F\) across that interval. For finite endpoints \(a<b\) in such an interval, this gives \(\int_a^b f(x)\,dx=F(b)-F(a)\). Taking one-sided limits gives the corresponding identity up to a jump location; on the tails, the assumed improper-integral condition gives the identity using the limits 0 and 1.

Add these continuous increases over all the intervals and add the jumps at \(x_1,\ldots,x_k\). Each intermediate endpoint value cancels: the value after one interval is followed by the jump, if any, and then by the next interval’s increase. The sum therefore telescopes from \(F(-\infty)=0\) to \(F(\infty)=1\). The continuous contributions combine to \(\int_{-\infty}^{\infty}f(x)\,dx\), and each jump contributes \(F(x_i)-F(x_i-)\). This proves the identity. \(\square\)

With these assumptions, the recovered components describe the whole law: probability at each jump location is the jump size, and probability away from those atoms is obtained by integrating \(f\). Equivalently, for any Borel set \(A\),

The conditions matter. A CDF that is merely differentiable between jumps need not be absolutely continuous there; differentiability alone does not guarantee that integrating its derivative recovers the full increase. For the piecewise CDFs considered here, confirm that the stated pieces are absolutely continuous before interpreting their derivatives as the complete continuous component.

Common Mistakes and Practical Checks

A frequent error is to treat the CDF value at a point as the atom size. It is the difference between the value after the jump and the left limit that gives the atom. For example, in the first worked example \(F(1)=0.7\), but the atom at 1 is \(0.7-0.5=0.2\), not \(0.7\). Another error is to call the derivative at a point the probability of that point. The derivative describes continuous probability per unit length; its integral over an interval gives probability there. A point’s atom instead comes from a jump.

When checking a proposed decomposition, verify both the local and global conditions. Locally, jump sizes and densities must be nonnegative, and the CDF must not decrease between pieces. Globally, the CDF must start at limit zero, end at limit one, and the sum of all atom sizes and density integrals must equal one. A mismatch signals a missed jump, an incorrect endpoint limit, a derivative or integration error, or an invalid CDF.