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For an intrinsic probability density function $\mathcal{P}(q)$, the law of total probability averages the conditional density from part c. An object observed with ratio $Q$ can only have $q\le Q$, soThis is an Abel transform of $\mathcal{P}(q)/\sqrt{1-q^2}$. Its normalization follows by reversing the order of integration and using ${\int }_q^1\mathcal{P}(Q\mid q)dQ=1$.

Withthe factor $\sqrt{1-q^2}$ cancels from the Abel transform:HenceThis probability density function is normalized and again favors rounder projections.

For an ordinary intrinsic density, inversion of the Abel transform givesIf $\mathcal{P}(Q)=1$, the integral is $\pi /2$ for every $q>0$, so its derivative vanishes. The missing probability is an endpoint atom: all systems must be infinitely thin,Indeed, putting $q=0$ directly into part c gives $\mathcal{P}(Q\mid 0)=1$. Thus a uniform apparent-axis-ratio distribution corresponds to a Dirac delta distribution of ideal zero-thickness disks, rather than to a regular intrinsic density.