Jointly distributed random variables

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. 

Continuing to use the data in Exercise 1, consider, once again, only the nonzero observations. Suppose that the sampling mechanism is as follows y and another normally distributed random variable, z, have population correlation 0.7. The two variables, y and z are sampled jointly. When z is greater than zero, y is reported. When z is less than zero, both z and y are discarded. Exactly 35 draws were required in order to obtain the preceding sample. Estimate p and a. [Hint Use Theorem 20.4.]... 

Frequently the scalar 2 drms accuracy does not adequately describe the position accuracy and the distribution of the error vector provides useful additional information. The probability density of two jointly normal random variables is given by... 

For emphasis, aU of these results about estimators arrive fi om the formulation of a sample as a collection of independent and identically distributed random variables. Their joint distribution is derived from the -fold product of the density function of the underlying distribution. Once these ideas are expressed as multiple integrals, deriving results in statistics are exercises in calculus and analysis. 

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... 

Now consider N pairs of random variables each having the distribution above, and each pair being statistically independent of all other pairs. Define w = 2 -t wn> = 2 -i and define HN(r,t) to be the joint moment generating function of w,z. 

An important concept is the marginal density function which will be better explained with the joint bivariate distribution of the two random variables X and Y and its density fXY(x, y). The marginal density function fxM(x) is the density function for X calculated upon integration of Y over its whole range of variation. If X and Y are defined over SR2, we get... 

The simplest of these models which permits a detailed discussion of the decay of correlations is a random walk model in which a set of random walkers whose positions are initially correlated is allowed to diffuse the motion of any single random walker being independent of any other member of the set. Let us assume that there are r particles in the set and motion occurs on a discrete lattice. The state of the system is, therefore, completely specified by the probabilities Pr(nlf n2,..., nr /), (tij = — 1, 0, 1, 2,. ..) in which Pr(n t) is the joint probability that particle 1 is at n1( particle 2 is at n2, etc., at time l. We will also use the notation Nj(t) for the random variable that is the position of random walker j at time t. Reduced probability distributions can be defined in terms of the Pr(n t) by summation. We will use the notation P nh, rth,..., ntj I) to denote the distribution of random walkers iu i2,..., i at time t. We define... 

A physical system S that evolves probabilistically in time can be mathematically described by a time-dependent random variable X(t). It is assumed that (1) one can measure values xlt x2, x3,.. . , xn of X(t) at instants ti,t2,ti,...,tn representing the possible states of the system S and (2) one can define a set of joint probability distribution functions... 

The PPK approach estimates the joint distribution of population specific pharmacokinetic model parameters for a given drug. Fixed effect parameters quantify the relationship e.g. of clearance to individual physiology like function of liver, kidney, or heart. The volume of distribution is typically related to body size. Random effect parameters quantify the inter-subject variability which remains after the fixed effects have been taken into account. Then the observed concentrations will still be randomly distributed around the concentration time course predicted by the model for an individual subject. This last error term called residual variability... 

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... 

The variable x in the preceding formulas denotes a quantity that varies. In our context, it signifies a reference value. If the variable by chance may take any one of a specified set of values, we use the term variate (i.e, a random variable). In this section, we consider distributions of single variates (i.e., univariate distributions). In a later section, we also discuss the joint distribution of two or more variates bivariate or multivariate distributions). 

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. 

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... 

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... 

An extension to the bivariate normal distribution is when there are more than two random variables under consideration and their joint distribution follows a multivariate normal (MVN) distribution. The pdf for the MVN distribution with p random variables can be written as... 

Suppose the joint distribution of the two random variables x andy is... 

An example of a Bayesian network is given in fig. 13.8. The concentrations of the five species are random variables each depends on the concentration of its parents, but not on other species. The parent of C is B, the parents of B are A and E. The connectivity of network structure in fig. 13.8 states that there is a probability P(A) that the concentration of A has a given value a probability P(B A,E) that B has a given value given the concentrations of A and E, the parents and similarly for P(C B), P(D A), and P(E). The joint probability distribution, with the cited restrictions, is a product of these distributions, that is,... 

For the analysis of a gene network, gene expression is taken to be a probabilistic process and the level of gene expression of each gene is a random variable. The object of the analysis is the calculation of the joint probability distribution over the set of genes, not necessarily of all genes at a time but of a limited number, and to estimate its structure. There may be different but equivalent graphs that represent the observed distribution. 

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