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Now, the importance sampling problem then focuses on finding a biasing density such that the variance of the importance sampling estimator is less than the variance of the general Monte Carlo estimate. For some biasing density function, which minimizes the variance, and under certain conditions reduces it to zero, it is called an optimal ...
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. Mathematically, density is defined as mass divided by volume: [1] where ρ is the density, m is the mass, and V is the volume.
The unusual density curve and lower density of ice than of water is essential for much of the life on earth—if water were most dense at the freezing point, then in winter the cooling at the surface would lead to convective mixing. Once 0 °C are reached, the water body would freeze from the bottom up, and all life in it would be killed. [35]
Relative density ( ) or specific gravity ( ) is a dimensionless quantity, as it is the ratio of either densities or weights where is relative density, is the density of the substance being measured, and is the density of the reference. (By convention , the Greek letter rho, denotes density.) The reference material can be indicated using ...
Extensive properties. An extensive property is a physical quantity whose value is proportional to the size of the system it describes, [8] or to the quantity of matter in the system. For example, the mass of a sample is an extensive quantity; it depends on the amount of substance. The related intensive quantity is the density which is ...
Bulk density. In materials science, bulk density, also called apparent density, is a material property defined as the mass of the many particles of the material divided by the bulk volume. Bulk volume is defined as the total volume the particles occupy, including particle's own volume, inter-particle void volume, and the particles' internal ...
The density of states plays an important role in the kinetic theory of solids. The product of the density of states and the probability distribution function is the number of occupied states per unit volume at a given energy for a system in thermal equilibrium. This value is widely used to investigate various physical properties of matter.
Probability theory. In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is The parameter is the mean or expectation of the distribution (and also its median and mode), while ...