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At fixed intrinsic ratio $q$, part a givesSince $\mu$ is uniform, the change-of-variables formula for a probability density yields the conditional probability density functionThe inverse-square-root singularity at $Q=q$ is integrable, and direct integration gives one.

The printed values appear to favor model 2 because $-12.529<336.381$. They are not directly comparable: model 2 reports the Gaussian likelihood of $Z_i=\log T_i$, whereas model 3 reports a density for $T_i$. The change-of-variables formula for a probability density givesso the transformed model's AIC on the original response scale isBecause the standardized predictors have zero sample means and the ordinary-least-squares residuals sum to zero,Thereforewhich is slightly worse than model 3's $336.381$.

For an unoriented symmetry axis chosen uniformly on the sphere, $\cos i$ is uniform on $[0,1]$. At fixed intrinsic ratio $q$, the change-of-variables formula for a probability density gives