Simultaneous distribution functions example

There are many ways we could assign probability distribution functions to the increments N(t + sk) — N(t + tk) and simultaneously satisfy the independent increment requirement expressed by Eq. (3-237) however, if we require a few additional properties, it is possible to show that the only possible probability density assignment is the Poisson process assignment defined by Eq. (3-231). One example of such additional requirements is the following50... 

The three distribution functions for the three possible directions in space must be interdependent when the number of bonds N is small. For N = 1, for example. Lx + lJ + Lz = must hold, where b is the bond length. The three components of the overall distribution function become less dependent on each other with increasing number of bonds in the chain molecule. If is very large, and L is simultaneously much smaller than the square of the length of the fully stretched chain molecule, then the components can be considered to be independent of each other. The total probability is simply the product of the individual probabilities, i.e.,... 

The spectrometer provides two series of data as a function of time the number of detected ions and, simultaneously, a physical value indicating the mass of these ions. This can be, for example, the value of the magnetic field measured by a Hall probe in the case of a magnetic instrument. The ions of every mass appear with a certain distribution over a time period, as shown in the sketch. The surface under the curve is proportional to the number of detected ions, while the value of the magnetic field at the centroid of the peak is an indicator of the ion mass. This centroid and mass must therefore be determined. In order to achieve... 

Amalgams function as electron donors in the same manner as a mercury electrode, and in many cases a similar product distribution is obtained by direct electrolysis at a mercury electrode and by an amalgam reduction. However, simultaneously with the electron transfer through the double layer a transfer of cations takes place, and this may influence the reaction, for example, by ion-pair formation. 

The CCN behavior of ambient particles can be measured by drawing an air sample into an instrument in which the particles are subjected to a known supersaturation, a so-called CCN counter (Nenes et al. 2001). If the size distribution and chemical composition of the ambient particles are simultaneously measured, then the measured CCN behavior can be compared to that predicted by Kohler theory on the basis of their size and composition. Such a comparison can be termed a CCN closure, that is, an assessment of the extent to which measured CCN activation can be predicted theoretically [see, for example, VanReken et al. (2003), Ghan et al. (2006), and Rissman et al. (2006)]. The next level of evaluation is an aerosol-cloud drop closure, in which a cloud parcel model, which predicts cloud drop concentration using observed ambient aerosol concentration, size distribution, cloud updraft velocity, and thermodynamic state, is evaluated against direct airborne measurements of cloud droplet number concentration as a function of altitude above cloud base. The predicted activation behavior can also be evaluated by independent measurements by a CCN instrument on board the aircraft. Such an aerosol-cloud drop closure was carried out by Conant et al. (2004) for warm cumulus clouds in Florida. 

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