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to see this, consider the joint probability density function of formula_77.
to see this, consider the joint probability density function of formula_90.

to see this, consider the joint probability density function of "x"=("x"1,…,"x""n").
to see this, consider the joint probability density function of "x"=("x"1...,"x""n").

the collection of likelihood ratios formula_64 is a minimal sufficient statistic if formula_65 is discrete or has a density function.
the collection of likelihood ratios formula_77 is a minimal sufficient statistic if formula_78 is discrete or has a density function.

phd thesis, upc, barcelonabarris, j., & garcia-almirall, p. (2011) a density function of the appraisal value.
phd thesis, upc, barcelona*barris, j., & garcia-almirall, p. (2011) "a density function of the appraisal value".

thus, the conditional probability distribution is::formula_58with the first equality by definition of conditional probability density, the second by the remark above, the third by the equality proven above, and the fourth by simplification.
thus, the conditional probability distribution is::formula_71with the first equality by definition of conditional probability density, the second by the remark above, the third by the equality proven above, and the fourth by simplification.

if a particle is placed at formula_83 at time formula_84, then the probability density function associated with the position vector of the particle formula_85 will be the following:: formula_86which is a (multivariate) normal distribution evolving in time.
if a particle is placed at r = 0 at time "t" = 0, then the probability density function associated with the position vector of the particle r will be the following:: formula_92which is a (multivariate) normal distribution evolving in time.

for instance, the solution to the following problem:formula_76:formula_77:formula_78:formula_79is given by:formula_80== applications ===== particle diffusion ===one can model particle diffusion by an equation involving either:* the volumetric concentration of particles, denoted "c", in the case of collective diffusion of a large number of particles, or* the probability density function associated with the position of a single particle, denoted "p".
it is described by laplace's equation::formula_89== applications ===== particle diffusion ===one can model particle diffusion by an equation involving either:* the volumetric concentration of particles, denoted "c", in the case of collective diffusion of a large number of particles, or* the probability density function associated with the position of a single particle, denoted "p".