Cumulative distributions

Figure Bl.10.1. Poisson distribution for fi(left vertical axis). Cumulative Poisson distribution for fi= l(right vertical axis). The cumulative distribution is the sum of the values of the distribution from 0 to n, where n =, 2, 3, 4, 5 on this graph.

Fig. 1. Cumulative distribution function of half squared distances of amino acid pairs at residue distance d (d = 1,..., 20), truncated at Cmax = 72 (12Acutoff)...

For the robust estimation of the pair potentials, some obstacles had to be overcome. There are a huge number of different triples (si, Sk,i — k), and to find densities, we needed a way to group them in a natural way together into suitable classes. A look at the cumulative distribution functions (cdf s) of the half squared distances Cjfc at residue distance d = i — k (w.l.o.g. >0), displayed in Figure 1, shows that the residue distances 8 and higher behave very similarly so in a first step we truncated all residue distances larger than 8 to 8. 

Median Diameter. The median droplet diameter is the diameter that divides the spray into two equal portions by number, length, surface area, or volume. Median diameters may be easily determined from cumulative distribution curves. 

Let us consider a product which is sold entirely on the basis of personal recommendation. The rate of sale will depend on the number of people who have already bought the product. Thus initially sales will increase exponentially. Eventually the market will be saturated, and only replacement purchases will be made. If the frequency cui ve may be assumed to be symmetrical about a single maximum value, the cumulative distribution cui ve is known as the logistics cui ve and is defined by Eq. (9-91) ... 

A common form of societal risk measure is an F-N curve, which is normally presented as a cumulative distribution plot of frequency F... 

Figure 4.6 Shape of the Cumulative Distribution Function (CDF) for an arbitrary normal distribution with varying standard deviation (adapted from Carter, 1986)...

Complementary cumulative distribution for latent cancer fatalities. 

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... 

This is illustrated in Fig. 12.11. As the integral in Eq. (12.3) cannot be evaluated by elementary methods, the cumulative distribution function is determined from tables. 

To deal with all kinds of normal distributions of different means and variances, the cumulative distribution is further normalized. This introduces a new variable u = x - ix)/(t. This operation changes a N(p, a) distribution to a N(0, 1) distribution. From Eq. (12.3) the following is obtained ... 

If a measurement is repeated only a few times, the estimate for the distribution variance calculated from this sample is uncertain and the tiornial distribution cannot be applied. In this case another distribution is used, f his distribution is Student s distribution or the /-distribution, and it has one more parameter the number of degrees of freedom, t>. The /-distribution takes into account, through the p parameter, the uncertainty of the variance. The values of the cumulative /-distribution function cannot be evaluated by elementary methods, and tabulated values or other calculation methods have to be used. 

Anotlier fimction used to describe tlie probability distribution of a random variable X is tlie cumulative distribution function (cdf). If f(x) specifies tlie pdf of a random variable X, tlien F(x) is used to specify the cdf For both discrete and continuous random variables, tlie cdf of X is defined by ... 

For any distribution, the cumulative hazard function and the cumulative distribution junction are connected by a simple relationship. The probability scale for the cumulative distribution function appears on the horizontal axis at the top of hazard paper and is determined from that relationship. Thus, the line fitted to data on hazard paper... 

Positive Step Changes and the Cumulative Distribution. Residence time distributions can also be measured by applying a positive step change to the inlet of the reactor Cm = Cout = 0 for r<0 and C = Co for r>0. Then the outlet response, F i) = CouMICq, gives the cumulative distribution function. ... 

The differential distribution is related to the cumulative distribution and to the washout function by... 

Material flowing at a position less than r has a residence time less than t because the velocity will be higher closer to the centerline. Thus, F(r) = F t) gives the fraction of material leaving the reactor with a residence time less that t where Equation (15.31) relates to r to t. F i) satisfies the definition. Equation (15.3), of a cumulative distribution function. Integrate Equation (15.30) to get F r). Then solve Equation (15.31) for r and substitute the result to replace r with t. When the velocity profile is parabolic, the equations become... 

F r) Cumulative distribution function expressed in terms of tube radius for a monotonic velocity profile 15.29... 

Fig. 2 Particle size distributions, (by volume) of the impeller-batch (top) and the micro-mixer-continuous-flow (bottom) processes when manufacturing the pigment Yellow 12. The cumulative distributions are given as well [11]...

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