Probability density definition

Equation (3-104) (sometimes called the stationarity property of a probability density function) follows from the definition of the joint distribution function upon making the change of variable t = t + r... 

It is often important to be able to extend our present notion of conditional probability to the case where the conditioning event has probability zero. An example of such a situation arises when we observe a time function X and ask the question, given that the value of X at some instant is x, what is the probability that the value of X r seconds in the future will be in the interval [a,6] As long as the first order probability density of X does not have a Dirac delta function at point x, P X(t) = x = 0 and our present definition of conditional probability is inapplicable. (The reader should verify that the definition, Eq. (3-159), reduces to the indeterminate form in this case.)... 

We must now verify that the definition (3-219) does indeed generate a possible set of probability density functions. First of all, pXtfn, as defined by Eq. (3-219) is everywhere non-negative, and, moreover,... 

The aforementioned interpretation of (x,f) 2 as a probability density is possible because positive definite and (b) when integrated over all space, i.e., J i//(x,t) 2d3x, is time independent. (By a suitable normalization of (0 > this integral can always be made equal to 1.)... 

As different sources are considered, the statistical properties of the emitted field changes. A random variable x is usually characterized by its probability density distribution function, P x). This function allows for the definition of the various statistical moments such as the average. 

By definition, the average or expectation value of x is just the sum over all possible values of x of the product of x and the probability of obtaining that value. Since x is a continuous variable, we replace the probability by the probability density and the sum by an integral to obtain... 

The statistical fundamentals of the definition of CV and LD are illustrated by Fig. 7.8 showing a quasi-three-dimensional representation of the relationship between measured values and analytical values which is characterized by a calibration straight line y = a + bx and their two-sided confidence limits and, in addition (in z-direction) the probability density function the measured values. 

If we will consider arbitrary random process, then for this process the conditional probability density W xn,tn x, t, ... x i,f i) depends on x1 X2,..., x . This leads to definite temporal connexity of the process, to existence of strong aftereffect, and, finally, to more precise reflection of peculiarities of real smooth processes. However, mathematical analysis of such processes becomes significantly sophisticated, up to complete impossibility of their deep and detailed analysis. Because of this reason, some tradeoff models of random processes are of interest, which are simple in analysis and at the same time correctly and satisfactory describe real processes. Such processes, having wide dissemination and recognition, are Markov processes. Markov process is a mathematical idealization. It utilizes the assumption that noise affecting the system is white (i.e., has constant spectrum for all frequencies). Real processes may be substituted by a Markov process when the spectrum of real noise is much wider than all characteristic frequencies of the system. 

If the probability density wj(t,x0) of the first passage time of boundaries c and d exists, then by the definition [18] we obtain... 

It is easy to check that the normalization condition is satisfied at such a definition, wT(t.xoj dt = 1. The condition of nonnegativity of the probability density wx(f,x0) > 0 is, actually, the monotonic condition of the probability Q(t, xq). In the case where c and d are absorbing boundaries the probability density of transition time coincides with the probability density of the first passage time wT(t,x0y. 

Finally, for additional support of the correctness and practical usefulness of the above-presented definition of moments of transition time, we would like to mention the duality of MTT and MFPT. If one considers the symmetric potential, such that (—oo) = <1>( I oo) = +oo, and obtains moments of transition time over the point of symmetry, one will see that they absolutely coincide with the corresponding moments of the first passage time if the absorbing boundary is located at the point of symmetry as well (this is what we call the principle of conformity [70]). Therefore, it follows that the probability density (5.2) coincides with the probability density of the first passage time wT(f,xo) w-/(t,xo), but one can easily ensure that it is so, solving the FPE numerically. The proof of the principle of conformity is given in the appendix. 

The state of polarization is determined by the pair of complex numbers e and e2 the quantities ei 2 and e2 2 represent probability densities of a definite (linear or circular) polarization of the photon as determined by the unit vectors Xi and x2- Since ej and e2 are related by the normalization condition... 

I function which carries maximum information about that system. Definition of the -function itself, depends on a probability aggregate or quantum-mechanical ensemble. The mechanical state of the systems of this ensemble cannot be defined more precisely than by stating the -function. It follows that the same -function and hence the same mechanical state must be assumed for all systems of the quantum-mechanical ensemble. A second major difference between classical and quantum states is that the -function that describes the quantum-mechanical ensemble is not a probability density, but a probability amplitude. By comparison the probability density for coordinates q is... 

In this book, an alternative description based on the joint probability density function (PDF) of the species concentrations will be developed. (Exact definitions of the joint PDF and related quantities are given in Chapter 3.) The RTD function is in fact the PDF of the fluid-element ages as they leave the reactor. The relationship between the PDF description and the RTD function can be made transparent by defining a fictitious chemical species... 

Figure 4.2 Definition of the percentile p the curve is the probability density function f(x) of the continuous random variable X. xp is the pth percentile when the surface up to xp represents p percent of the total surface S under the curve.

As has been mentioned above, the inclusion of basis functions (49) with high power values, nik, is very essential for the calculations of molecular systems. It is especially important for highly vibrationally excited states where there are many highly localized peaks in the nuclear correlation function. To illustrate this point, we calculated this correlation function (it corresponds to the internuclear distance, r -p = r ), which is the same as the probability density of pseudoparticle 1. The definition of this quantity is as follows ... 

Parameter Two distinct definitions for parameter are used. In the first usage (preferred), parameter refers to the constants characterizing the probability density function or cumulative distribution function of a random variable. For example, if the random variable W is known to be normally distributed with mean p and standard deviation o, the constants p and o are called parameters. In the second usage, parameter can be a constant or an independent variable in a mathematical equation or model. For example, in the equation Z = X + 2Y, the independent variables (X, Y) and the constant (2) are all parameters. 

This I(S) is always defined when there is at least one distribution for which. S is true but it need not be finite. Thus, if the domain of definition of. S is the set of probability densities p(x) on the whole x axis, a trivial 5 (i.e., true for every p(x)) and also an S which merely gives the value of the first moment, has I(S) = — oo. On the other hand, if S states that the second moment is close to zero, /( S ) is very large, and I(S) -> + oo as this moment approaches zero. Of course there is no p(x) having a zero second moment (only a point distribution, which is not a p(x)). Thus it might seem natural to define I(S) as + oo when S defines an empty set. Then every S without exception has a unique I(S). 

For definiteness consider a closed, isolated physical system. If at t = 0 the quantity Y has the precise value y0 the probability density P(y, t) is initially 5(y — y0). It will tend to Pe(y) as t increases. If y0 is macroscopically different from the equilibrium value of Y it means that y0 is far outside the width of Pe(y), because macroscopically observed values are large compared to the equilibrium fluctuations. We also know from experience that the fluctuations remain small during the whole process. That means that P y, t), for each t, is a sharply peaked function of y. The location of this peak is a fairly well-defined number, having an uncertainty of the order of the width of the peak, and is to be identified with the macroscopic value y(t). For definiteness one customarily adopts the more precise definition... 

Wavefunctions of electrons in atoms are called atomic orbitals. The name was chosen to suggest something less definite than an orbit of an electron around a nucleus and to take into account the wave nature of the electron. The mathematical expressions for atomic orbitals—which are obtained as solutions of the Schrodinger equation—are more complicated than the sine functions for the particle in a box, but their essential features are quite simple. Moreover, we must never lose sight of their interpretation, that the square of a wavefunction tells us the probability density of an electron at each point. To visualize this probability density, we can think of a cloud centered on the nucleus. The density of the cloud at each point represents the probability of finding an electron there. Denser regions of the cloud therefore represent locations where the electron is more likely to be found. 

Next, the definition of PSD is discussed based on the uncertainty regarding the size of the particle that is selected when a particle is selected. The original PSD q0 (l/l) is regarded as a probability density function, and the PSD satisfies the following standardization condition ... 

As it was already written above, we would like to study structural changes in the charge distribution between macroscopic objects, that is caused by the image forces, and depends on the wall-to-wall distance. To obtain direct structural information about the system, we will introduce a configurational analogue of the phase-space distribution function. At equilibrium, the definition of an fth order distribution function given by Eq. (12) can be applied to the equilibrium probability density [Eq. (13)], and the integration with respect to impulses can easily be carried out. We write for the rth order local density... 

In the last equations A(n) is the Helmholtz free energy of the total NVT system fully defined by the vector n = nm, n 2,... providing the molecular number in each tiny volume and iq) is the chemical potential at a given iq position, i.e., within the corresponding tiny volume. Note that the molecular number can be used as a continuous variable, given the fact that for any thermodynamic property in a macroscopic system the variation due to a single molecule is virtually equivalent to a differential. From the definition of the chemical potential and probability density in the r) space p(t, ti), we readily have... 

To provide a definition of the density matrix in terms of fundamental wave-functions first consider the generalization of the expectation value from quantum mechanics to quantum statistical mechanics. In the quantum statistical case, an additional average over the probability density needs to be considered in the calculation of the expectation value ... 

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