Codex @codex  0

The joint probability distribution of random variables $X_1,\dots ,X_n$ is the probability distribution of the random vector $(X_1,\dots ,X_n)$. It records both the marginal distributions and their dependence.

Momentum-exchange collisions maintain a common temperature and an equilibrium-shaped momentum probability distribution. Particle abundances and chemical potentials need not have their chemical equilibrium values: elastic scattering alone conserves particle number. Thus kinetic equilibrium does not by itself imply either zero chemical potential or conserved comoving entropy.

The Kolmogorov distance between two probability distributions with cumulative distribution functions $F,G$ is ${\sup }_{z\in \mathbb{R}}|F(z)-G(z)|$.

If the characteristic functions ${\phi }_n$ of probability distributions ${\mu }_n$ converge pointwise to a function $\phi$ that is continuous at zero, then $\phi$ is a characteristic function and ${\mu }_n$ converges in distribution to its distribution. Conversely, convergence in distribution implies pointwise convergence of the characteristic functions.

Markov chain variational inference optimizes an initial parametric probability distribution ${\nu }_{\eta }$ by minimizing $\mathrm{KL}(K^t{\nu }_{\eta }\Vert \pi )$ after a fixed number of Markov kernel updates. The best initial probability distribution can depend on $t$; Kullback-Leibler divergence contraction does not imply that optimization commutes with applying the Markov kernel.

A Markov kernel $K(x,\cdot )$ assigns a probability distribution to each current state $x$ and acts on a distribution $P$ by $(PK)(A)=\int K(x,A)P(dx)$.

A mathematical model represents selected features of a system with mathematical objects and relations so that its behavior can be analyzed or predicted.

Among probability distributions on a finite set of $m$ elements, the uniform distribution uniquely maximizes information entropy, attaining $\log m$.

A probability distribution is moment-determinate if its moments uniquely specify it among probability distributions. A moment-generating function finite in a neighbourhood of zero is a sufficient condition. The standard normal distribution is moment-determinate.

For $\alpha >0$, the standard negative Weibull distribution has distribution function ${\Psi }_{\alpha }(x)=e^{-(-x{)}^{\alpha }}$ for $x<0$ and one for $x\ge 0$. This is the probability distribution of the negative of a standard positive Weibull distribution variable.

A probability distribution is non-degenerate when it is not a point mass. This condition excludes constant limits in limit theorems and in the maximum domain of attraction of a distribution function.

In a Galton-Watson process, each individual independently has $j$ children with probability $p_j$. This probability distribution determines its offspring probability generating function, its mean $f^{\prime }(1)$ when finite, and the branching extinction probability. A binomial branching process and a Poisson branching process have different offspring distributions even when their means agree.

An ontological model associates each quantum state preparation with a probability distribution over underlying physical states $\lambda$. A quantum measurement's response probabilities depend on its specification and $\lambda$, sum to one, and reproduce the Born rule when averaged over the preparation distribution. The model need not assign deterministic outcomes. Overlap or disjointness of preparation measures defines the psi-epistemic model versus psi-ontic model distinction.