Then the probability of the measurement outcome lying in an interval B of R is | E < sub > A </ sub >( B ) ψ |< sup > 2 </ sup >.

Then the probability of a measurement returning the value λ < sub > i </ sub > can be calculated by integrating the spectral measure

Then ( Ω, F, P ) is a probability space, with sample space Ω, event space F and probability measure P.

Then the number of particles passing perpendicularly through unit area of a cross-section per unit time is the probability flux ;

Then the probability of getting this observation ( of being in interval j ) is approximately

Then has probability density function.

Let be the mean of the values in associated with class c, and let be the variance of the values in associated with class c. Then, the probability of some value given a class,, can be computed by plugging into the equation for a Normal distribution parameterized by and.

Then the probability of failure is given by

Then they are independent, but not necessarily identically distributed, and their joint probability distribution is given by the Bapat – Beg theorem.

Then, the burden shifts to the plaintiff, to affirmatively present evidence to show that they have a reasonable probability of prevailing on the action.

Then there is some probability distribution m on the interval and some random variable Y such that

Then with probability 1:

Then the theory is invariant under the Poincaré group if for every ray Ψ of the Hilbert space and every group element ( a, L ) is given a transformed ray Ψ ( a, L ) and the transition probability is unchanged by the transformation:

Then the phase space distribution determines the probability that the system will be found in the infinitesimal phase space volume.

Then the probability density function for is given by

Then the Wishart distribution is the probability distribution of the p × p random matrix

Then, if n ≥ p, has a Wishart distribution with n degrees of freedom if it has a probability density function given by

Then the conditional probability given is a function such that is the conditional expectation of the indicator function for A:

Then μ is a probability measure on the σ-algebra of Borel subsets of I.

Then for all non-uniform probabilistic polynomial time algorithms that output and of increasing length k, the probability that and is a negligible function in k.

Then, Boltzmann's entropy formula gives for the probability distribution function, A = const.

Then converges to with probability 1 as.

Then participants were asked to evaluate the probability of her being a feminist, the probability of her being a bank teller, or the probability of being both a bank teller and feminist.

Then g ( E ) is called the density of states if the energy spectrum is continuous.

Then it was pointed out that " seeing the most submarines " depended not only on the number of submarines present, but also on the number of eyes looking ; i. e., patrol density.

Then the density matrix for this system is

Then the corresponding density operator is:

Then, if we take the partial trace over the environment, we'd find the density state is approximately described by

Then the densities of the sums of two, three, and four independent identically distributed variables, each having the original density, are shown in the following figures.

Then the Radon – Nikodym theorem provides the function g, equal to the density of μ with respect to Q.

Then the photon flux density ( watts per metre squared usually ) of the transmitted or reflected light is measured with a photodiode, charge coupled device or other light sensor.

Then, the set of primes v of K not in S such that F < sub > v </ sub > ⊆ X has density

Then measuring can yield a result anywhere in the interval ( a, b ), with probability density function ; i. e., a result between y and z will occur with probability

Let H and G be Hilbert spaces of dimension n and m respectively, and Φ be a quantum operation taking the density matrices acting on H to those acting on G. Then there are matrices

Then it was installed a stage and 400 graduates of the police auxiliary entered into the field, which were located on the elements of turf protection, it was obtained a density of four people per square meter, which is the density presented at massive events.

Then the total population of the urban areas ( or localities ) in Sweden was 7, 631, 952 on an area of 5, 286. 23 km², which gives an average population density of 1, 444 / km².

Then north and south of Chaves population density increases ( 68 per km² ).

Then, upon substitution the assumed relation between pressure and density into we obtain the equation

Then our choice function can choose the least element of every set under our unusual ordering.

Then an arithmetic function a is

Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.

Then is a group whose identity element is The group inverse of an arbitrary group element is the function inverse

Then, using the periodic Bernoulli function P < sub > n </ sub > defined above and repeating the argument on the interval, one can obtain an expression of ƒ ( 1 ).

Then a mathematical procedure is used to find a filter transfer function which can be realized ( within some constraints ) and which approximates the desired response to within some criterion.

Then an " estimator " is a function that maps the sample space to a set of sample estimates.

Let R be a domain and f a Euclidean function on R. Then:

For instance, suppose that each input is an integer z in the range 0 to N − 1, and the output must be an integer h in the range 0 to n − 1, where N is much larger than n. Then the hash function could be h

Then ƒ is invertible if there exists a function g with domain Y and range X, with the property:

Then for a specific value x of X, the function L ( θ | x )

Then,, and the constant function turns into the identity by substitution.

Then, in 1905, to explain the photoelectric effect ( 1839 ), i. e., that shining light on certain materials can function to eject electrons from the material, Albert Einstein postulated, based on Planck ’ s quantum hypothesis, that light itself consists of individual quantum particles, which later came to be called photons ( 1926 ).

The algorithm for deciding this is conceptually simple: it constructs ( the description of ) a new program t taking an argument n which ( 1 ) first executes program a on input i ( both a and i being hard-coded into the definition of t ), and ( 2 ) then returns the square of n. If a ( i ) runs forever, then t will never get to step ( 2 ), regardless of n. Then clearly, t is a function for computing squares if and only if step ( 1 ) terminates.

Then the output is related to the input by the transfer function as

Then because the construction of f was uniform, this is a recursive function of two variables.

Then, its electrical field traces out a helical pattern as a function of time.

Then the function

Then the function

Then, recalling that the likelihood function is defined up to a multiplicative constant, it is just as valid to say that the likelihood function is approximately

Then, on considering the lengths of the intervals to decrease to zero, the likelihood function for an observation from the discrete component is