The probability density function

Figure 6.6 The probability density function and the expectation curve...

Fig. 10.19 The probability density of the extreme value distribution typical of the MSP scores for random sequena The probability that a random variable with this distribution has a score of at least x is given by 1 - exp[-e -where u is the characteristic value and A is the decay constant. The figure shows the probability density function (which corresponds to the function s first derivative) for u = 0 and A = 1.

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. 

In the introduction to this section, two differences between "classical" and Bayes statistics were mentioned. One of these was the Bayes treatment of failure rate and demand probttbility as random variables. This subsection provides a simple illustration of a Bayes treatment for calculating the confidence interval for demand probability. The direct approach taken here uses the binomial distribution (equation 2.4-7) for the probability density function (pdf). If p is the probability of failure on demand, then the confidence nr that p is less than p is given by equation 2.6-30. 

The probability density function of u is shown for four points in Fig. 11.16, two points in the wall jet and two points in the boundary layer close to the floor. For the points in the wall jet (Fig. 11.16<2) the probability (unction shows a preferred value of u showing that the flow has a well-defined mean velocity and that the velocity is fluctuating around this mean value. Close to the floor near the separation at x/H = I (Fig. 11.16f ) it is hard to find any preferred value of u, which shows that the flow is irregular and unstable with no well-defined mean velocity and large turbulent intensity. From Figs. 11.15 and 11.16 we can see that LES gives us information about the nature of the turbulent fluctuations that can be important for thermal comfort. This type of information is not available from traditional CFD using models. 

The thermally produced noise voltage X(t) appearing across the terminals of a hot resistor is often modeled by assuming that the probability density function for X(t) is gaussian,... 

The probability density function pY can differentiation with the result... 

The next example will illustrate the technique of calculating moments when the probability density function contains Dirac delta functions. The mean of the Poisson distribution, Eq. (3-29), is given by... 

Here we have a case where all, not only most, of the area under the probability density function is located within V 2 standard deviations of the mean, but where this fact alone gives a very misleading picture of the arcsine distribution, whose area is mainly concentrated at the edges of the distribution. Quantitatively, this is borne out by the easily verified fact that one half of the area is located outside of the interval [—0.9,0.9]. 

The last example brings out very clearly that knowledge of only the mean and variance of a distribution is often not sufficient to tell us much about the shape of the probability density function. In order to partially alleviate this difficulty, one sometimes tries to specify additional parameters or attributes of the distribution. One of the most important of these is the notion of the modality of the distribution, which is defined to be the number of distinct maxima of the probability density function. The usefulness of this concept is brought out by the observation that a unimodal distribution (such as the gaussian) will tend to have its area concentrated about the location of the maximum, thus guaranteeing that the mean and variance will be fairly reasdnable measures of the center and spread of the distribution. Conversely, if it is known that a distribution is multimodal (has more than one... 

The characteristic function is thus seen to be the Fourier transform17 of the probability density function p+. The fact that the function etv< is bounded, e<0< = 1, implies that the characteristic function of a distribution function always exists and, moreover, that... 

One of the most important properties of Fourier transforms and, consequently, of characteristic functions, is their invertibility. Given a characteristic function M, one can calculate the probability density function p by means of the inversion formula... 

This last step is not by far as trivial as it sounds and requires a fairly involved argument to establish it on a rigorous basis.44 Moreover, in the absence of any additional assumptions about the distribution function of the individual summands, it is not possible to conclude that the probability density function of s approaches (1/V27r)e"l2/. This subtlety is not apparent in our argument but shows up when an attempt is made to give a careful discussion of the last step in the proof. 

The probability density function pY,fa ( m) is the m-dimensional Fourier transform of Eq. (3-259) but, once again, this can only be evaluated explicitly in certain special cases. 

Although we cannot easily obtain expressions for the probability density functions of Y(t), it is a simple matter to calculate its various moments. We shall illustrate this technique by calculating all possible first and second moments of Y(t) i.e., E[7(t)] and E[Y(t)Y(t + r)], — oo < v < oo. The pertinent characteristic function for this task is MYQt (hereafter abbreviated MYt) given by... 

We shall conclude this section by investigating the very interesting behavior of the probability density functions of Y(t) for large values of the parameter n. First of all, we note that both the mean and the covariance of Y(t) increase linearly with n. Roughly speaking, this means that the center of any particular finite-order probability density function of Y(t) moves further and further away from the origin as n increases and that the area under the density function is less and less concentrated at the center. For this reason, it is more convenient to study the normalized function Y ... 

To describe single-point measurements of a random process, we use the first-order probability density function p/(/). Then p/(/) df is the probability that a measurement will return a result between / and / -I- df. We can characterize a random process by its moments. The nth moment is the ensemble average of /", denoted (/"). For example, the mean is given by the first moment of the probability density function. 

The shape of the probability density function, depends on the system. Some examples are shown in Fig. 4-4. This figure also contains probability density of age (see Section 4.2.3). Figure 4-4a might correspond to a lake with inlet and outlet on opposite sides of the lake. Most water molecules will then have a residence time in the lake roughly equal to the time it takes for the mean current to carry the water from the... 

X 10 years old, this implies that the content of the reservoir today is about half of what it was when the Earth was formed. The probability density function of residence time of the uranium atoms originally present is an exponential decay function. The average residence time is 6.5 x 10 years. (The average value of... 

Since one is only rarely interested in the density at a precise point on the z-axis, the cumulative probability (cumulative frequency) tables are more important in effect, the integral from -oo to +z over the probability density function for various z > 0 is tabulated again a few entries are given in Fig. 1.13. 

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.)... 

The distribution of stretches can be quantified in terms of the probability density function F (X)=dN(X)/dX, where dN(A) is the number of points that have values of stretching between A and (X+dX) at the end of period n. Another possibility is to focus on the distribution of log A. In this case we define the measure //n(logA)=dA(logA)/d(logA). 

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. 

Here, the systems 0 and 1 are described by the potential energy functions, /0(x), and /i(x), respectively. Generalization to conditions in which systems 0 and 1 are at two different temperatures is straightforward. 1 and / i are equal to (/cbTqJ and (/ i 7 i j, respectively. / nfxj is the probability density function of finding system 0 in the microstate defined by positions x of the particles ... 

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