A mixture distribution first chooses a component $j$ with nonnegative mixture weights $w_j$ summing to one, then samples from its law $P_j$. Consequently $P(A)={\sum }_j{w}_j{P}_j(A)$, and any finite moment-generating function is the same weighted sum of the component transforms. A component may be a Dirac measure, so a mixture may have both discrete and continuous parts.

For a positive random variable $X$ with finite nonzero expected value $\mu$, its integrated tail law has density $f_I(x)=\mathbb{P}(X>x)/\mu$ on $x\ge 0$. The tail integral formula for moments normalizes it. For $r>0$ its exponential transform is $\int e^{rx}{f}_I(x)dx=[M_X(r)-1]/(\mu r)$, with extended-value interpretation when needed.

For a real random variable $X$ with mean $\mu =\mathbb{E}X$, finite positive variance $0<{\sigma }^2=\mathrm{Var}(X)<\infty$, and finite fourth moment, kurtosis is the fourth central moment divided by the square of the variance:The standard score $Z$ measures distance from the mean in standard deviations. Its fourth power gives a value ten standard deviations away a weight $10^4=10000$ inside the expected value. Thus a small probability mass far from the mean can strongly affect kurtosis. It is useful for describing the contribution of extreme standard scores, but does not by itself describe the height or shape of a probability density function near its peak.^[ref]

The normal distribution has kurtosis $3$. A random scale can increase kurtosis even when the conditional distribution is normal. A constant random variable has zero variance, so this ratio is undefined. If the variance is finite and positive but the fourth central moment is infinite, the ratio can be written $+\infty$ in an extended sense; it is not a finite kurtosis. If the variance itself is infinite or the mean is undefined, the displayed normalization is unavailable.

For a probability measure $\mu$ on a topological space, its support isA point belongs to the support when observations can fall arbitrarily close to it with positive probability. This is a neighbourhood condition, not a requirement that the point itself have positive probability mass or that a chosen density be positive there.

For the uniform distribution on $(0,1)$ viewed as a law on the real line, the support is $[0,1]$, including both endpoints. A density $f(x)=3x^2/2$ on $[-1,1]$ vanishes at zero, but zero still lies in its support because every open neighbourhood of zero has a positive integral of $f$. A discrete distribution with mass $2^{-n}$ at $1/n$, $n\ge 1$, likewise has zero in its support despite assigning it probability mass zero.

If a distribution's support equals an interval, every open subinterval of that interval has positive probability. The weaker condition $\mu (I)=1$ does not imply that the support equals $I$: a uniform law on $(0,1)$ also assigns probability one to $[0,2]$, but assigns zero to its subinterval $(3/2,7/4)$. This distinction matters when using support to describe feasible observations or randomized effort in a contest.

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.

For an additive arithmetic function $f$, $N^{-1}{\sum }_{n\le N}|f(n)-A_f(N){|}^2\ll B_f(N{)}^2$, where $A_f$ is the mean of an additive arithmetic function and $B_f(N{)}^2={\sum }_{p^k\le N}|f(p^k){|}^2/p^k$. The implied constant is absolute.

The multiplication table problem asks how many distinct integers occur as $ab$ with $1\le a,b\le N$. Counting the $N^2$ pairs greatly overcounts the distinct products.

The Cramér model selects integers $n\ge 3$ independently with probability $1/\log n$, and conventionally sets $U_1=0$, $U_2=1$. It is a random model for the distribution of primes, not an assertion of independence for actual prime numbers.

For a real strongly additive arithmetic function with uniformly bounded values on primes, put $A(x)={\sum }_{p\le x}f(p)/p$ and $B(x{)}^2={\sum }_{p\le x}f(p{)}^2/p$. If $B(x)\to \infty$, then $(f(n)-A(N))/B(N)$ under uniform sampling from $[N]$ converges to the standard normal distribution. For the prime omega function, the mean and variance scales are $\log \log N$.

The dimensionless spectrum is the contribution to a statistically isotropic field variance per logarithmic wavenumber interval: $⟨X^2⟩=\int d\log k{\Delta }_X(k)$, with any necessary smoothing or regulators. Some conventions denote this same quantity ${\Delta }_X^2$. The definition, not the superscript convention, fixes its meaning.

Chemical equilibrium of $n+{\nu }_e\leftrightarrow p+e^{-}$, with negligible electron chemical potential, gives the stated neutron-to-proton ratio. Positive electron-neutrino degeneracy reduces the neutron fraction, whereas negative degeneracy increases it. The primordial helium mass fraction then responds through neutron-limited helium synthesis while protons outnumber neutrons; a sufficiently neutron-rich mixture instead becomes proton limited, so that trend must not be extrapolated indefinitely. Extra neutrino energy density also accelerates cosmic expansion, changing weak freeze-out and the time available for neutron decay. These effects can compete; a quantitative Big Bang nucleosynthesis calculation must account for the altered reaction rates as well as the altered expansion rate.

Search over the finite interval $0\le y\le b$ for the least witness to a primitive recursive predicate, with an explicitly chosen default if there is none. The result is a primitive recursive function, by scanning this finite interval using primitive recursion. This differs from unbounded minimization, which may fail to terminate.

For a fixed effective machine coding, the predicate that program $e$ on input $x$ halts within $t$ steps is primitive recursive. Code finite configurations, iterate a total single-step function by primitive recursion, and test for the halt state. This supplies a bounded simulation even though the unrestricted halting problem is undecidable.

The multiplicative arithmetic function $f(n)=3^{\omega (n)}$ counts assignments of three labels to each distinct prime factor. It also counts pairs of squarefree divisors whose least common multiple is the radical of an integer $n$.

In the local rest frame of a gas, the pressure tensor is the directional momentum flux $P_{ij}=\int p_i{v}_{j}f(\mathbf{p})d^{3}p$, where $f$ is the phase-space distribution function per unit momentum volume and $\mathbf{v}$ is particle velocity. For an isotropic distribution, $P_{ij}=P{\delta }_{ij}$, recovering the kinetic pressure of an isotropic gas. An anisotropic distribution can exert different normal pressures along different directions.

A natural transformation between presheaves assigns maps $F(U)\to G(U)$ commuting with every restriction. For presheaves of abelian groups or modules, those maps preserve the specified algebraic structure.

For an ideal projective measurement with the Lüders rule, condition on both an initial quantum state preparation and a final successful postselection. Its outcome probabilities are proportional to $|⟨b|P_j|a⟩{|}^2$, where $a$ is the forward-evolved initial state and $b$ the backward-evolved final state. The normalization must be nonzero. The expression is symmetric in these two boundary states, although its derivation uses the ordinary Born rule and conditional state update.

A linear map between ordered vector spaces is positive if it carries the positive cone of the domain into the positive cone of the codomain. For function spaces this means $f\ge 0⟹Tf\ge 0$. A Feller semigroup consists of such positive maps on continuous functions.

For an integer base $p\ge 2$, a nonnegative integer has a finite positional representation with digits $k_i\in \{0,\dots ,p-1\}$. Repeated Euclidean division gives the digits uniquely; appending zero high-place digits has no effect. The base need not be prime for uniqueness, but a prime base is useful in Lucas theorem and modular binomial coefficients.

A hereditary base representation expands not only the number in base $b$, but also every exponent recursively in that same base. Changing the base at every depth defines the Goodstein base-change operation; replacing it by omega defines an ordinal rank of a Goodstein term.