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The support is $x>\theta$ and the probability density function isThe change of variables $z=\log (x-\theta )$ proves normalization. The threshold $\theta$ is a lower endpoint, distinct from the log-mean $m$ and log-standard-deviation $\sigma$. In measured sea-ice pressure ridge spacings, finite ridge widths and detection rules can create such a lower endpoint.

The square of a uniform distribution variable on $(0,1)$ has cumulative distribution function $\sqrt{x}$ and an integrable inverse-square-root probability density function. Reflection replaces $x$ by $1-x$ in that density.

A statistical functional assigns a target value to each probability measure in a statistical model. Examples include an expected value, a quantile, and an integral of a power of a probability density function. Its behavior along statistical paths determines whether first-order influence-function representers are available.

Surrogate Hamiltonian Monte Carlo uses an auxiliary smooth probability density function $\nu$ to generate trajectories but applies a Metropolis–Hastings acceptance probability using the intended target $\mu$. For Gaussian momenta and an involutive Metropolis proposal generated by flipped leapfrog integration, the acceptance probability is $\min (1,\mu (x^{\prime })e^{-\Vert p^{\prime }{\Vert }^2/2}/[\mu (x)e^{-\Vert p{\Vert }^2/2}])$. Only the surrogate gradient is required during the trajectory; target density evaluations are still required at the endpoints.

In the classical risk model with premium rate $c>0$ and claim Poisson process rate $\lambda$, the first-claim decomposition for survival probability implies the displayed equation for ultimate survival probability. Change the first-claim integral to an integral over available capital and differentiate its lower limit. The bounded survival probability function convolved with the integrable claim probability density function is continuous, so this also establishes the needed differentiability. In particular ${\phi }^{\prime }(0)=(\lambda /c)\phi (0)$.

For nonoverlapping triangular sea-ice pressure ridges with a common along-track side slope $\tan \delta$, a ridge whose peak exceeds $h$ contributes $2\cot \delta dh$ of track length in a draft interval $dh$. Summing over peaks proves the identity. Above the cutoff of an exponential ridge-draft model, $g(h)=A{e}^{-bh}$ with $A=2B/(b\tan \delta )$. This draft occupation is not the normalized peak probability density function; level ice and shallower parts supply the remaining probability.

Variational inference approximates a posterior distribution by minimizing a Kullback-Leibler divergence over a chosen family of probability distributions. For an unnormalized posterior density $h$ and a trial probability density function $q$, minimizing ${\mathbb{E}}_q[\log q-\log h]$ is equivalent to minimizing $\mathrm{KL}(q\Vert h/\int h)$.

For an absolutely continuous distribution function with positive probability density function near its right endpoint of a distribution, write $\eta =f/(1-F)$ for its hazard function. The limits $t\eta (t)\to \alpha$ at an infinite endpoint or $(x_F-t)\eta (t)\to \alpha$ at a finite endpoint imply attraction to the Fréchet distribution or negative Weibull distribution, respectively. A continuously differentiable reciprocal hazard function with $(1/\eta {)}^{\prime }\to 0$ implies attraction to the Gumbel distribution, provided the survival function tends to zero at the endpoint.

For a probability density function $w$ on a domain, a weighted Poincare inequality has the formThe optimal $\lambda$ is the spectral gap of the associated weighted diffusion operator.