Joint and Marginal Distribution

Frequently, more than one random variable is associated with an outcome. For example, an insurance file provides information on a customer s auto and homeowner policy. Vital signs monitored on a hospital patient include the systolic and diastolic blood pressure. The relationship between several variables can be investigated through their joint probability distribution. 

To illustrate jointly distributed r.v. s, we consider data on grades (A = 4, B = 3, C = 2, D = l)ina probability and a statistics course for a sample of 200 Northwestern University undergraduate engineering smdents. A tabulation of these grades is shown in Table 2.5. Suppose the transcript of one of these smdents is drawn at random. What is the probability that the student received an A in probability and a B in statistics  

TABLE 2.6 Joint Distribution of Probability and Statistics Grades 

This probability is simply 22 the proportion of students with this particular 

Similarly, the joint p.d.f. of two continuous bivariate random vectors (X, F) is denoted by/(X,y), which satisfies 

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This is simple to illustrate with an example. From the sun and rain example What is the probability that it will rain on the second day given it rained on the first day From the above equation and using the joint and marginal distributions... 

In the case that the joint or marginal distribution of the test statistics is unknown, p-values can be estimated by resampling methods such as permutation and bootstrap. For example, consider a permutation algorithm to estimate p-values with large biological data in the following manner. First, permute the N sample columns of the data matrix and compute test statistics for each biomarker candidate. Let tij, be test statistics for the th permutation. When repeating this procedure many limes (e.g., B = 100 times), the permutation p-value for hypothesis Hj is... 

Limited Information Maximum Likelihood Estimation). Consider a bivariate distribution for x and y that is a function of two parameters, a and fi The joint density is j x,y a,p). We consider maximum likelihood estimation of the two parameters. The full information maximum likelihood estimator is the now familiar maximum likelihood estimator of the two parameters. Now, suppose that we can factor the joint distribution as done in Exercise 3, but in this case, we have, fix,y a, ft) — f(y x.a.f )f(x a). That is, the conditional density for y is a function of both parameters, but the marginal distribution for x involves only... 

Finally, we adopt a notation involving conditional averages to express several of the important results. This notation is standard in other fields (Resnick, 2001), not without precedent in statistical mechanics (Febowitz et al, 1967), and particularly useful here. The joint probability P A, B) of events A and B may be expressed as P A, B) = P A B)P B) where P B) is the marginal distribution, and P A B) is the distribution of A conditional on B, provided that P B) 0. The expectation of A conditional on B is A B, the expectation of A evaluated with the distribution P(A B) for specified B. In many texts (Resnick, 2001), that object is denoted as E(A B) but the bracket notation for average is firmly established in the present subject so we follow that precedent despite the widespread recognition of a notation (A B) for a different object in quantum mechanics texts. 

The joint distribution for a first-order Markov chain depends only on the one-step transition probabilities and on the marginal distribution for the initial state of the process. This is because of the Markov property. A first-order Markov chain can be fit to a sample of realizations from the chain by fitting the log-linear (or a nonlinear mixed effects) model to [To, Li, , YtiYt] for T realizations because association is only present between pairs of adjacent, or consecutive, states. This model states that the odds ratios describing the association between To and Yt are the same at any combination of states at the time points 2,..., T, for instance. 

Figure A.4 Joint pdf of the bivariate normal distribution and the marginal distributions for X and Y. In this example, both X and Y have a standard normal distribution with a correlation between variables of 0.6.

Notice that the conditional pdf is normally distributed wherever the joint pdf is sliced parallel to X. In contrast, the marginal distribution of Y or X evaluates the pdf across all of X or Y, respectively. Hence, the mean and variance of the marginal and condition distribution from a bivariate normal distribution can be expressed as ... 

Having established that the null hypothesis of joint taste parameter distribution is supported in the data, we eoneentrated on the model without socio-economic variables (MXLC) and used the parameter estimates of the joint distribution of tastes to simulate taste segment shares, and marginal and conditional distributions of WTP for each of the three production related attributes. These are reported in Table 3. 

Example 2.14 Covariance from Probability and Statistics Grades. The joint distribution of the probability and statistics grades, X and Y, is given later in Table 2.6 and their marginal distributions are given also later in Table 2.7. What is the covariance of X and F ... 

The r.v. s Xj, X2,..., X. are said to be mutually independent if and only if their joint distribution factors into the product of their marginal distributions, that is,... 

Note that the item-based performance measures / , and depend only on the marginal distribution of Xi for each i, while the product-based performance measures / and B depend on the joint distribution of Xi, i = 1,. . . , m. 

The marginal distribution for a specific variable Xt in a Bayesian Network can be computed from the joint distribution (see e.g. Jensen, 2001). In the example in Fig. 1 we may e.g. he interested in the marginal distribution of Z, and can compute this by marginaUsing outX and Y as follows ... 

A challenging problem is to establish the distribution G, the joint distribution of the values of aU the component failure rates. Here independence is assumed, as is usual in practice, and only marginal distributions are assigned to each individual component failure rate. 

We win conclude this paper by stating again that Bayesian framework is very appropriate for dealing with uncertainty in industrial practice. The more powerful feature of Bayesian uncertainty setting is that a unique joint distribution exists for observable and unobservable variables. Working with conditional or marginal distributions allows to isolate or aggregate respectively uncertainty sources. 

This survey of IT probes of chemical bonds continues with some rudiments on the entropic characteristics of the dependent probability distributions and information descriptors of a transmission of signals in communication systems [3,4,7,8]. For two mutually dependent (discrete) probability vectors of two separate sets of events a and b, P(a) = P(fl) =Pi) =p and P(b) = P(pp = qj = q, one decomposes the joint probabilities of the simultaneous events a/ b = [a,Air) into these two schemes, P afdb) = P(fli/ b) = Jt,y] = 31, as products of the margin probabilities of events in one set, say a, and the corresponding conditional probabilities P( la) = [P(/li) = 3t,y/Pj] of outcomes in set b, given that events a have already occurred [jty=p (j i). The relevant normalization conditions for the joint and conditional probabilities then read ... 

The joint distribution is not known, but can be determined approximately from the marginal distributions fyj (Y) and fs (s) and the correlation coefficients py< s by utilizing Morgensterns 2 or Natafs transormatlon ". ... 

We are treating the parameter as a nuisance parameter, and marginalizing it out of the joint posterior. One of the advantages of the Bayesian approach is that we have a clear-cut way to find the predictive distribution that works in all circumstances. 

We then marginalize the mean pj out of the joint distribution. We note that only the first term contains pj, and it has the form of a normal density. Hence the marginal distribution ofyj given the parameter r is given by... 

Remark 3 It is worth mentioning that although the independence of input variables is crucial for the applicability of MMA, and thus for Subset Simulation, they need not be identically distributed. In other words, instead of Eq. 18, the joint PDF 7r( ) can have a more general form, = Y[k= ik kixk), where n (-) is the marginal distributions of x which is not necessarily Gaussian. In this case, the expression for the acceptance ratio in Eq. 26 must be replaced by = ... 

Bonded-bolted joints have good load distribution and are generally designed so that the bolts take all the load. Then, the bolts would take all the load after the bond breaks (because the bolts do not receive load until the bond slips). The bond provides a change in failure mode and a sizable margin against fatigue failure. 

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... 

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