For a Gaussian response system ( or a simple RC roll off ), the rise time is approximated by:

For general matrices, Gaussian elimination is usually considered to be stable in practice if you use partial pivoting as described below, even though there are examples for which it is unstable.

For example, if one assumes that data arise from a univariate Gaussian distribution, then one has assumed a Gaussian model:.

For example, a mixture of Gaussians with one Gaussian at each data point is dense is the space of distributions.

For example, a common choice for A < sub > o </ sub > is a Gaussian wave packet:

For example, a Gaussian wavefunction ψ might take the form:

For an ideal single-mode Gaussian beam, the D4σ, D86 and 1 / e < sup > 2 </ sup > width measurements would give the same value.

For both families, the lowest-order solution describes a Gaussian beam, while higher-order solutions describe higher-order transverse modes in an optical resonator.

For a Gaussian beam propagating in free space, the spot size ( radius ) w ( z ) will be at a minimum value w < sub > 0 </ sub > at one place along the beam axis, known as the beam waist.

For a Gaussian beam, the BPP is the product of the beam's divergence and waist size.

For a Gaussian intensity ( i. e. power density, W / m < sup > 2 </ sup >) distribution in a single-mode optical fiber, the mode field diameter is that at which the electric and magnetic field strengths are reduced to 1 / e of their maximum values, i. e., the diameter at which power density is reduced to 1 / e < sup > 2 </ sup > of the maximum power density, because the power density is proportional to the square of the field strength.

For large numbers the Poisson distribution approaches a normal distribution, typically making shot noise in actual observations indistinguishable from true Gaussian noise except when the elementary events ( photons, electrons, etc.

For example, the inverse of cumulative Gaussian distribution

For sufficiently nice prior probabilities, the Bernstein-von Mises theorem gives that in the limit of infinite trials and the posterior converges to a Gaussian distribution independent of the initial prior under some conditions firstly outlined and rigorously proven by Joseph Leo Doob in 1948, namely if the random variable in consideration has a finite probability space.

For any affine transformation of the Gaussian plane, z mapping to a z + b, a ≠ 0, a triangle is transformed but does not change its shape.

For a Gaussian distribution this is the best unbiased estimator ( that is, it has the lowest MSE among all unbiased estimators ), but not, say, for a uniform distribution.

For example, for a laser producing pulses with a Gaussian temporal shape, the minimum possible pulse duration Δt is given by

For the HeNe laser with a 1. 5 GHz spectral width, the shortest Gaussian pulse consistent with this spectral width would be around 300 picoseconds ; for the 128 THz bandwidth Ti: sapphire laser, this spectral width would be only 3. 4 femtoseconds.

For large α, the shape of the Kaiser window ( in both time and frequency domain ) tends to a Gaussian curve.

For the Gaussian random approximation, the structure function of Tatarski ( 1961 ) can be described in terms

For example, the Gaussian curvature of a cylindrical tube is zero, the same as for the " unrolled " tube ( which is flat ).

* For an orthogonal parametrization ( i. e., ), Gaussian curvature is:

* For a surface described as graph of a function, Gaussian curvature is:

* For a normal distribution, the arithmetic mean is equal to both the median and the mode, other measures of central tendency.

For example, as in pin and as in spin are allophones for the phoneme in the English language because they cannot distinguish words ( in fact, they occur in complementary distribution ).

For most players, many calls ( bids, doubles and redoubles, and sometimes even passes ) are not made with the intention that they become the final contract, but to describe the strength and distribution of the player's hand, so that the partnership can reach an informed conclusion on their best contract, and / or to obstruct the opponents ' bidding.

For example, the traditional common law rule through most of the 19th century was that a plaintiff could not recover for a defendant's negligent production or distribution of a harmful instrumentality unless the two were in privity of contract.

For every real number x, the cumulative distribution function of a real-valued random variable X is given by

For example, the Riemann hypothesis is a conjecture from number theory that ( amongst other things ) makes predictions about the distribution of prime numbers.

For example, if n samples are taken from a Cauchy distribution, one may calculate the sample mean as:

For certain types of point sets, such as a uniform random distribution, by intelligently picking the splitting lines the expected time can be reduced to O ( n log log n ) while still maintaining worst-case performance.

For example, J. B. Clark saw the co-ordinating function in production and distribution as being served by entrepreneurs ; Frank Knight introduced managers who co-ordinate using their own money ( financial capital ) and the financial capital of others.

For example, the illegal manufacture, distribution or possession of controlled substances may be a felony, although possession of small amounts may be only a misdemeanor.

For the next two years, Hoover worked 14-hour days from London, administering the distribution of over two million tons of food to nine million war victims.

For most types of hashing functions the choice of the function depends strongly on the nature of the input data, and their probability distribution in the intended application.

For instance, if we impose some requirements on a distribution f, we can attempt to translate these requirements in terms of the Fourier transform of f. The Paley – Wiener theorem is an example of this.

For a symmetric distribution ( where the median equals the midhinge, the average of the first and third quartiles ), half the IQR equals the median absolute deviation ( MAD ).

For some discussion of the subtleties of the Eötvös experiment, such as the local mass distribution around the experimental site ( including a quip about the mass of Eötvös himself ), see Franklin.

For easier distribution of large programs, multiple class files may be packaged together in a. jar file ( short for Java archive ).

For a probability density function f ( x ) with the cumulative distribution function F ( x ), the Lorenz curve L ( F ( x )) is given by:

For a cumulative distribution function F ( x ) with inverse x ( F ), the Lorenz curve L ( F ) is given by:

For a continuous charge distribution in motion, the Lorentz force equation becomes:

For some people, the obvious solution to this concern is to redefine the wage structure politically to achieve a socially preferable distribution of income.

For a probability distribution, the mean is equal to the sum or integral over every possible value weighted by the probability of that value.

For example, mainline churches, with their upper class congregations, " promote order, stability, and conservatism, and in so doing proved to be a powerful source of legitimation of the status quo and of existing disparities in the distribution of wealth and power " because much of the wealth of the church comes from the congregation.

For example, the Internet includes web sites, blogs, podcasts, and various other technologies built on top of the general distribution network.

For example, this map of Europe has been distorted to show population distribution, while the rough shape of the continent is still discernable.

For the special case where is a joint probability distribution and the loss function is the negative log likelihood a risk minimization algorithm is said to perform generative training, because can be regarded as a generative model that explains how the data were generated.