Probability density distribution Normal

LDA is the first classification technique introduced into multivariate analysis by Fisher (1936). It is a probabilistic parametric technique, that is, it is based on the estimation of multivariate probability density fimc-tions, which are entirely described by a minimum number of parameters means, variances, and covariances, like in the case of the well-knovm univariate normal distribution. LDA is based on the hypotheses that the probability density distributions are multivariate normal and that the dispersion is the same for all the categories. This means that the variance-covariance matrix is the same for all of the categories, while the centroids are different (different location). In the case of two variables, the probability density fimction is bell-shaped and its elliptic section lines correspond to equal probability density values and to the same Mahala-nobis distance from the centroid (see Fig. 2.15A). 

Quadratic discriminant analysis (QDA) is a probabilistic parametric classification technique which represents an evolution of EDA for nonlinear class separations. Also QDA, like EDA, is based on the hypothesis that the probability density distributions are multivariate normal but, in this case, the dispersion is not the same for all of the categories. It follows that the categories differ for the position of their centroid and also for the variance-covariance matrix (different location and dispersion), as it is represented in Fig. 2.16A. Consequently, the ellipses of different categories differ not only for their position in the plane but also for eccentricity and axis orientation (Geisser, 1964). By coimecting the intersection points of each couple of corresponding ellipses (at the same Mahalanobis distance from the respective centroids), a parabolic delimiter is identified (see Fig. 2.16B). The name quadratic discriminant analysis is derived from this feature. 

The probability density distribution function p(t) expressed by Eq. (1.13) is shown in Figure 1.1. This distribution is well known as the normal distribution or Gaussian distribution. 

In the chemical engineering field, there are quite a few cases in which the value of variance is discussed. For a significant discussion on the value of variance, the probability density distribution function should have the same form. Additionally, from the viewpoint of information entropy, it can be understood that the discussion based on the value of variance becomes significant when the probability density function/distribution is that for the normal distribution. 

According to the central limit theorem, the probability density distribution of x for a sample of size n will be given by a standard normal distribution with the test statistic defined [2] as... 

Figure 5.9 is a schematic comparison between the Normal (Gaussian) and Weibull distributions for plane strain fracture toughness Kic of a material in terms of the respective probability density distributions, The minimum and char-... 

The quantum mechanical state n) has no direct physical interpretation, but its absolute square, Y p=Y Y , can be interpreted as a probability density distribution. This soBorn interpretation implies for a single particle that the wave function has to be normalized, i.e., integration over all dynamical variables of a system must yield unity. 

We will call c the concentration, although it is a form of concentration density with respect to the index variable x. The function h(x) is a distribution function (much like a probability density function) normalized so that... 

Figure 4.3 Shapes of the probability density function (PDF) for the (a) normal, (b) lognormal and (c) Weibull distributions with varying parameters (adapted from Carter, 1986)...

The shape of the Normal distribution is shown in Figure 3 for an arbitrary mean, /i= 150 and varying standard deviation, ct. Notice it is symmetrical about the mean and that the area under each curve is equal representing a probability of one. The equation which describes the shape of a Normal distribution is called the Probability Density Function (PDF) and is usually represented by the term f x), or the function of A , where A is the variable of interest or variate. 

The probability density of the normal distribution f x) is not very useful in error analysis. It is better to use the integral of the probability density, which is the cumulative distribution function... 

Figure 1.11. The probability density of the normal distribution. Because of the symmetry often only the right half is tabulated.

Figure 1.14. The probability density functions for several f-distributions (/ = 1, 2, 5, resp. 100) are shown. The f-distribution for / = 100 already very closely matches a normal distribution.

In everyday analytical work it is improbable that a large number of repeat measurements is performed most likely one has to make do with less than 20 replications of any detemunation. No matter which statistical standards are adhered to, such numbers are considered to be small , and hence, the law of large numbers, that is the normal distribution, does not strictly apply. The /-distributions will have to be used the plural derives from the fact that the probability density functions vary systematically with the number of degrees of freedom,/. (Cf. Figs. 1.14 through 1.16.)... 

Figure 3.8. The transformation of a rectangular into a normal distribution. The rectangle at the lower left shows the probability density (idealized observed frequency of events) for a random generator versus x in the range 0 < jc < 1. The curve at the upper left is the cumulative probability CP versus deviation z function introduced in Section 1.2.1. At right, a normal distribution probability density PD is shown. The dotted line marked with an open square indicates the transformation for a random number smaller or equal to 0.5, the dot-dashed line starting from the filled square is for a random number larger than 0.5.

Figure 12. Potential energy contour plots for He + I Cl(B,v = 3) and the corresponding probability densities for the n = 0, 2, and 4 intermolecular vibrational levels, (a), (c), and (e) plotted as a function of intermolecular angle, 0 and distance, R. Modified with permission from Ref. 40. The I Cl(B,v = 2/) rotational product state distributions measured following excitation to n = 0, 2, and 4 within the He + I Cl(B,v = 3) potential are plotted as black squares in (b), (d), and (f), respectively. The populations are normalized so that their sum is unity. The l Cl(B,v = 2/) rotational product state distributions calculated by Gray and Wozny [101] for the vibrational predissociation of He I Cl(B,v = 3,n = 0,/ = 0) complexes are shown as open circles in panel (b). Modified with permission from Ref. [51].

The principle of Maximum Likelihood is that the spectrum, y(jc), is calculated with the highest probability to yield the observed spectrum g(x) after convolution with h x). Therefore, assumptions about the noise n x) are made. For instance, the noise in each data point i is random and additive with a normal or any other distribution (e.g. Poisson, skewed, exponential,...) and a standard deviation s,. In case of a normal distribution the residual e, = g, - g, = g, - (/ /i), in each data point should be normally distributed with a standard deviation j,. The probability that (J h)i represents the measurement g- is then given by the conditional probability density function Pig, f) ... 

If the mathematical model of the process under consideration is adequate, it is very reasonable to assume that the measured responses from the i,h experiment are normally distributed. In particular the joint probability density function conditional on the value of the parameters (k and ,) is of the form,... 

A time-independent wave function is a function of the position in space (r = x,y,z) and the spin degree of freedom, which can be either up or down. The physical interpretation of the wave function is due to Max Born (25, 26), who was the first to interpret the square of its magnitude, > /(r)p, as a probability density function, or probability distribution function. This probability distribution specifies the probability of finding the particle (here, the electron) at any chosen location in space (r) in an infinitesimal volume dV= dx dy dz around r. I lu probability of finding the electron at r is given by )/(r) Id V7, which is required to integrate to unity over all space (normalization condition). A many-electron system, such as a molecule, is described by a many-electron wave function lF(r, r, l .I -.-), which when squared gives the probability den-... 

Continuous distribution functions Some experiments, such as liquid chromatography or mass spectrometry, allow for the determination of continuous or quasi-continuous distribution functions, which are readily obtained by a transition from the discrete property variable X to the continuous variable X and the replacement of the discrete statistical weights g, by the continuous probability density g(X). For simplicity, we assume g(X) as being normalized J ° g(X)dX = 1. Averages and moments of a quantity Y(X) are defined by analogy to the discrete case as... 

A random p-dimensional vector x follows a /7-variate normal distribution, x Np(fi, ) if its probability density function is given by... 

The normal distribution, A Y/l, o 2), has a mean (expectation) fi and a standard deviation cr (variance tr2). Figure 1.8 (left) shows the probability density function of the normal distribution N(pb, tr2), and Figure 1.8 (right) the cumulative distribution function with the typical S-shape. A special case is the standard normal distribution, N(0, 1), with p =0 and standard deviation tr = 1. The normal distribution plays an important role in statistical testing. 

FIGURE 1.8 Probability density function (PDF) (left) and cumulative distribution function (right) of the normal distribution cr2) with mean /a and standard deviation cr. The quantile q defines a probability p. 

FIGURE 2.4 Probability density function of the uniform distribution (left), and the logit-transformed values as solid line and the standard normal distribution as dashed line (right). 

The figure shows U >. S L in this region and Da is predominantly small. At the highest Reynolds numbers the region is entered only for very intense turbulence, U > SL. The region has been considered a distributed reaction zone in which reactants and products are somewhat uniformly dispersed throughout the flame front. Reactions are still fast everywhere, so that unbumed mixture near the burned gas side of the flame is completely burned before it leaves what would be considered the flame front. An instantaneous temperature measurement in this flame would yield a normal probability density function—more importantly, one that is not bimodal. 

As is shown in Eqns. 2-48 and 2-49, the probability density W(e) of electron energy states in the reductant or oxidant particles is represented as a normal distribution function (Gaussian distribution) centered at the most probable electron level (See Fig. 2-39.) as expressed in Eqns. 8-10 and 8-11 ... 

If we examine the current geographical distribution of a mutation, it is hard to estimate the value of the population density n at the position and time where the mutation originates. It makes sense to treat n as a random variable selected from a certain probability density p n). The constraints imposed onp n) are the conservation of the normalization condition f p(n) dn = 1 and the range of variation, noc > n > 0. The maximum information entropy approach leads to a uniform distribution... 

---