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If $W$ has probability density function $f(w;q)$ and cumulative distribution function $F_W(w;q)$, then $T=\exp (\alpha +\sigma W)$, $\sigma >0$, has density $f((\log t-\alpha )/\sigma ;q)/(\sigma t)$ and distribution function $F_W((\log t-\alpha )/\sigma ;q)$ for $t>0$. The factor $1/(\sigma t)$ follows from the change-of-variables formula for a probability density. Upper truncation at $M$ divides the density by $F_W((\log M-\alpha )/\sigma ;q)$.

For the log-transformed location-scale model, let $F_W$ denote the distribution function associated with $f$. The change-of-variables formula for a probability density gives the untruncated density and distribution functionConsequently its upper-truncated likelihood is ${\prod }_{i}p(t_i)/F(M{)}^n$ on the observed support, and the profile likelihood maximizes this over $\alpha ,\sigma ,q$. Restricting to the corresponding subfamilies gives the gamma distribution and lognormal distribution profiles. Write the plotted likelihood ratio asIt measures how well each endpoint fits relative to that model's own best endpoint, after nuisance refitting. It is not a probability density function for $M$ and need not integrate to one. Endpoints whose ratios are near one are relatively compatible with the observations; a rapidly falling profile indicates more information favoring endpoints near the maximum.

All three profiles favor the smallest admissible endpoint, approximately 14 days; their information about larger endpoints differs substantially. Reading the actual PDF legend and curves, the dotted gamma distribution profile falls initially but is nearly flat around $0.60$ from about 18 days onward. The solid generalized log-gamma profile declines much further, to roughly $0.20$ at 19 days and $0.18$ at 21 days. The dashed lognormal distribution profile falls most strongly, to roughly $0.14$ at 18 days and $0.08$ at 21 days. These are graphical readings, not recomputed fits.

Thus the gamma distribution fit gives weak discrimination between moderate and very large endpoints: much of its untruncated mass can already be below these endpoints, making further changes to the truncation have little effect. Over the displayed range, the lognormal distribution fit gives stronger relative evidence against large endpoints, and the more flexible generalized log-gamma fit is intermediate. The solid curve remains close to the dashed curve near the maximum but has appreciably more relative support farther out. The plot shows slowing declines rather than a sharply determined finite upper endpoint; it cannot establish a biological maximum.

Normalization is essential to this comparison. Although the generalized family includes the two special families and therefore has at least as large a raw maximized likelihood at any fixed $M$, its normalization constant can also be larger. Its normalized profile need not lie above the normalized profiles of its submodels. Their individual maxima have all been replaced by one, so this plot alone cannot compare overall goodness of fit or select a family while accounting for extra fitted parameters.

A set $\{M:R(M)\ge c\}$ is a relative-likelihood support set once $c$ is chosen. Turning it into a calibrated confidence interval requires an appropriate sampling argument, simulation or other justified method. In particular, a usual Wilks theorem chi-squared cutoff cannot simply be assumed for this changing-support endpoint problem. The original incubation observations are not supplied here, so numerical refitting or calibrated limits cannot be recovered from the graph alone.

Here $\Lambda$ is planar Lebesgue measure. For a Borel set $B\subseteq [0,\infty )$, polar coordinates giveThe pushforward is therefore locally finite and atomless. By the mapping theorem for Poisson point processes, $f(\Pi )$ is a homogeneous Poisson process on $[0,\infty )$ with constant intensity $\pi$. This is the two-dimensional radial volume transform of a homogeneous Poisson point process.

Put $S_k=R_k^2$. The variable $S_k$ is the $k$th arrival time of that rate-$\pi$ Poisson process, hence has the gamma distribution with shape $k$ and rate $\pi$:Since $s=r^2$ and $ds/dr=2r$, the change-of-variables formula for a probability density gives the Kth-nearest-neighbour distance in a homogeneous Poisson point process

On the interior of the support $b<m$, differentiating the joint distribution of Brownian motion and its running maximum gives the joint probability densityMake the change-of-variables formula for a probability density $y=m$ and $z=m-b$. Its inverse is $m=y$, $b=y-z$, and its Jacobian determinant has absolute value one. Since $2m-b=y+z$, for $y,z\ge 0$ this yields

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

If $X\sim N(\mu ,{\sigma }^2)$ with $\sigma >0$, then $Z=(X-\mu )/\sigma$ has the standard normal distribution. The change-of-variables formula for a probability density gives $f_Z(z)=\sigma f_X(\mu +\sigma z)=\varphi (z)$, the standard normal density. Thus standardization of a normal random variable removes both parameters without changing the probabilities of corresponding events.