Joint distribution function

Joint Distribution Functions.—The purpose of this section is to extend the results obtained in previous sections to averages that are not necessarily of the form... 

All averages of the form (3-96) can be calculated in terms of a canonical set of averages called joint distribution functions by means of an extension of the theorem of averages proved in Section 3.3. To this end, we shall define the a order distribution function of X for time spacings rx < r2 < < by the equation,... 

The properties of joint distribution functions can be stated most easily in terms of their associated probability density functions. The n + mth order joint probability density function px. . , ( > ) is defined by the equation... 

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... 

The multidimensional theorem of averages can be used to calculate the higher-order joint distribution functions of derived sets of time functions, each of which is of the form... 

Another instructive example concerns the joint distribution function of the pair of time functions Zx(t) and Z2(t) defined by... 

We conclude this section by introducing some notation and terminology that are quite useful in discussions involving joint distribution functions. The distribution function F of a random variable associated with time increments fnf m is defined to be the first-order distribution function of the derived time function Z(t) = + fn),... 

A few minutes thought should convince the reader that all our previous results can be couched in the language of families of random variables and their joint distribution functions. Thus, the second-order distribution function FXtx is the same as the joint distribution function of the random variables and 2 defined by... 

In this connection, we shall often abuse our notation somewhat by referring to FXZx Ts as the joint distribution function of the random variables X(t + rx) and X(t + r2) instead of employing the more precise but cumbersome language used at the beginning of this paragraph. In the same vein, the distribution function FXJn.rym will be referred to loosely as the joint distribution function of the random variables X(t + rj),- -, X(t + r ), Y(t + ri), -,Y t + r m). 

Once again, it should be emphasized that the functional form of a set of random variables is important only insofar as it enables us to calculate their joint distribution function in terms of other known distribution functions. Once the joint distribution function of a group of random variables is known, no further reference to their fractional form is necessary in order to use the theorem of averages for the calculation of any time average of interest in connection with the given random variables. 

A random process can be (and often is) defined in terms of the random variable terminology introduced in Section 3.8. We include this alternate definition for completeness. Limiting ourselves to a single time function X( ), it is seen that X(t) is completely specified as a random process by the specification all possible finite-order joint distribution functions of the infinite set of random variables T, — oo < t < oo, defined by the equations... 

The Poisson process represents only one possible way of assigning joint distribution functions to the increments of counting functions however, in many problems, one can argue that the Poisson process is the most reasonable choice that can be made. For example, let us consider the stream of electrons flowing from cathode to plate in a vacuum tube, and let us further assume that the plate current is low enough so that the electrons do not interact with one another in the... 

Joint distribution functions, in terms of associated probability density functions, 133 notation, 143... 

Since independence of U and W is assumed, the joint distribution function fXy x, y) is... 

Exercise. Take v non-overlapping intervals and express the characteristic function G(k1, k2,..., kv) of the joint distribution function of their occupation numbers iVi, N2,Nv in terms of the... 

Exercise. Find the hierarchy of joint distribution functions y ,t ) (the /s and fs are integers) for the finite Markov chain defined by a given T and Pi(yi, 0). 

This observation has importance when we take into account the irreversibility. Due to irreversibility, the damped oscillator proceeds to thermal equilibrium with the thermal bath. This thermal equilibrium can be characterized in terms of classical statistic theory. However, in classical statistics, random variables have a joint distribution function, which could exist in the case of quantum theory if the operators are compatible. The commutator relation (Equation (100)) is compatible this physical picture, but from Equations (100) and (101), we obtain... 

Consider the configuration space distribution function P(r ), Eq. (5.6). Mathematically, it is the joint distribution function (see Section 1.5.2) to find the N particles of the system in their respective positions in configuration space, that is, P(r )[Pg.179]

If all the particles in the system are identical then r i and r2 can be the coordinates of any two particles in the system. It is sometimes convenient to use a normalization that will express the fact that, if we look at the corresponding neighborhoods of ri and r2, the probability to find these neighborhoods occupied by any tM>o particles increases in a statistically determined way with the number of particles in the system. This is achieved by multiplying the joint distribution function (r i, r2)... 

Noting that N(N — 1) is the total number of pairs in the system, represents the density of such pairs per unit volume. This concept can be generalized the reduced joint distribution function for particles 1,..., n is given by... 

We can measure and discuss z(Z) directly, keeping in mind that we will obtain different realizations (stochastic trajectories) of this function from different experiments performed imder identical conditions. Alternatively, we can characterize the process using the probability distributions associated with it. P(z, Z)random variable z at time Z is in the interval between z and z +- dz. P2(z2t2 zi fi )dzidz2 is the probability that z will have a value between zi and zi + dz at Zi and between Z2 and Z2 -F t/z2 at t, etc. The time evolution of the process, if recorded in times Zo, Zi, Z2, - - , Zn is most generally represented by the joint probability distribution Piz t , , z iUp. Note that any such joint distribution function can be expressed as a reduced higher-order function, for example. 

Notice that two assumptions have been made normality of the responses and constant variance. The result is that the conditional distribution itself is normally distributed with mean 0O + (fix and variance normal distribution. Also, any conditional probability distribution function of Y has the same standard deviation after scaling the resulting probability distribution function to have an area of 1. 

Consider two statistic variables, say x andy, for each of which a distribution function, say f x) and g x), and for both of which a joint distribution function, F x,y), is defined. We call the variables independent if and only if... 

In order to apply the concepts of probability theory to electron correlation, we have to consider the case where the two variables have the same distribution function. We have also to take into account that q( ) is not the distribution function for one electron but for n electrons, in the same way as Ji(ri, 2) is the joint distribution function for n(n — 1) pairs. The equation defining independent electrons therefore becomes... 

The joint distribution function P( A",>, t) is converted formally into a probability in function space, P([w(r)], t), which is a functional of w(r) (see van Kampen, 1981, pp. 346-7). From a mathematical point of view this procedure is badly founded, since the probability density in function space is not defined. 

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