Joint Distributions

In many experiments there will be more than a single random variable of interest, say Xj, X2, X3,. .. etc. These variables can be conceptualized as a k-dimen-sional vector that can assume values (xj, X2, X3. etc). For example, age, height, weight, sex, and drug clearance may be measured for each subject in a study or drug concentrations may be measured on many different occasions in the same subject. Joint distributions arise when there are two or more random variables on the same probability space. Like the one-dimensional case, a joint pdf is valid if 

In other words, if all the random variables are independent, the joint pdf can be factored into the product of the individual pdfs. For two random variables, Xi and X2, their joint distribution is determined from their joint cdf 

Many times, the density of only one of the random variates in a joint pdf is of interest, while the other variables are considered nuisance variables. The density of one random variable alone is called the marginal pdf and is obtained by integrating out (or averaging out) the 

Assume that X and Y are random variables that have a joint distribution that is bivariate normal. The joint pdf between X and Y is 

What the conditional pdf does is slice the joint pdf at X and evaluate the pdf at that point in space. 

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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... 

A first approach to the definition of the confidence regions in parameter space follows the linear approximation to the parameter joint distribution that we have already used If the estimates are approximately normally distributed around 9 with dispersion [U. U.] then an approximate 100(1 - a)%... 

Figure 1. Results of Monte Carlo simulations for 1500 pairs of x, y points with a mean of 100, Ganssian errors of 10 (1 o), and four different x-y error correlations (p). Elhpses show 95% confidence hmits for the joint x-y distribution. Note that the ellipses extend farther than the 2o range of either the x or j errors themselves—a non-intuitive characteristic of joint distributions that arises because an x- (or y-) value deviating less than expected permits ay- (orx-) value that deviates more than expected.

Involved joint distribution Symmetric Symmetric or asymmetric... 

Clustering lacks the strict Bayesian requirement that input patterns be identifiable with a known prototype for which underlying joint distributions or other statistical information is known. Rather, the clustering approach... 

Its derivation implies a succession of two formal procedures. First, it is necessary to color the homopolymer globule units marking every z -th unit by color of/ with the probability waj(rl) which coincides with the ratio of concentration of units Ma, at point rl to the overall concentration of all units at this point. As a result of such a coloring, the joint distribution for configurations and conformations of proteinlike heteropolymers is obtained. Integration of this distribution over coordinates of all units results in the desired molecular-structure distribution (Eq. 23). 

The problem of finding the joint distribution of macromolecules P(ti, ti2 mi, m2) for numbers of internal, ( i, 2). and external, (mi, m2), blocks can be easily solved by the statistical method [1,3]. This is possible because the succession of blocks in a macromolecule is described by the ab-... 

Assuming that the errors are uncorrelated, the joint distribution of the biases is the product of the individual biases, that is,... 

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

The confidence intervals defined for a single random variable become confidence regions for jointly distributed random variables. In the case of a multivariate normal distribution, the equation of the surface limiting the confidence region of the mean vector will now be shown to be an n-dimensional ellipsoid. Let us assume that X is a vector of n normally distributed variables with mean n-column vector p and covariance matrix Ex. A sample of m observations has a mean vector x and an n x n covariance matrix S. 

A probabilistic model will typically require distributions for multiple inputs. Therefore, it is necessary to consider the joint distribution of multiple variables as well as the individual distributions, i.e., we must address possible dependencies among variables. At least, we want to avoid combinations of model inputs that are unreasonable on scientific grounds, such as the basal metabolic rate of a hummingbird combined with the body weight of a duck. 

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