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This type of mixture, being a finite sum, is called a finite mixture, and in applications, an unqualified reference to a "mixture density" usually means a finite mixture. The case of a countably infinite set of components is covered formally by allowing n = ∞ {\displaystyle n=\infty \!} .
For example, a mixture of two normal distributions with different means may result in a density with two modes, which is not modeled by standard parametric distributions. Another example is given by the possibility of mixture distributions to model fatter tails than the basic Gaussian ones, so as to be a candidate for modeling more extreme events.
In materials science, a general rule of mixtures is a weighted mean used to predict various properties of a composite material. [ 1 ] [ 2 ] [ 3 ] It provides a theoretical upper- and lower-bound on properties such as the elastic modulus , ultimate tensile strength , thermal conductivity , and electrical conductivity . [ 3 ]
J i is the diffusion flux vector of the i th species (for example in mol/m 2-s), M i is the molar mass of the i th species, ρ is the mixture density (for example in kg/m 3). The is outside the gradient operator. This is because: =, where ρ si is the partial density of the i th species.
Figure 1. A simple bimodal distribution, in this case a mixture of two normal distributions with the same variance but different means. The figure shows the probability density function (p.d.f.), which is an equally-weighted average of the bell-shaped p.d.f.s of the two normal distributions.
In chemistry, the mass concentration ρ i (or γ i) is defined as the mass of a constituent m i divided by the volume of the mixture V. [1]= For a pure chemical the mass concentration equals its density (mass divided by volume); thus the mass concentration of a component in a mixture can be called the density of a component in a mixture.
Density (volumetric mass density or specific mass) is a substance's mass per unit of volume. The symbol most often used for density is ρ (the lower case Greek letter rho ), although the Latin letter D can also be used.
Because the individual masses of the ingredients of a mixture sum to , their mass fractions sum to unity: ∑ i = 1 n w i = 1. {\displaystyle \sum _{i=1}^{n}w_{i}=1.} Mass fraction can also be expressed, with a denominator of 100, as percentage by mass (in commercial contexts often called percentage by weight , abbreviated wt.% or % w/w ; see ...