Bivariate distribution

For long linear chains the second condition is supported by the Stockmayer bivariate distribution (8,9) which shows the bivariate distribution of chain length and composition is the product of both distributions, and the compositional distribution is given by the normal distribution whose variance is inversely proportional to chain length. 

Fig, 33.4. Nintety-five percent confidence limit for a bivariate distribution as class envelope. 

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... 

Fig. 33. Probability density function for Gaussian bivariate distribution...

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... 

Exercise. Compute the marginal and conditional probabilities for the following ring-shaped bivariate distribution ... 

Fig. 1. Cell cycle progression of V79 cells. Bivariate distributions of DNA content (jc-axis) vs BrdU uptake (y-axis) of V79 cells removed at hourly intervals after pulse labeling with BrdU. Cells in exponential growth, were pulse labeled with 10 pM BrdU for 20 min. After washing out the BrdU, the cells were resuspended in fresh medium and harvested at 1-h intervals. 

Fig. 2. FCM profiles of V79 cells 4 h after pulse labeling with BrtlU, The left-hand pane shows the bivariate distribution, the middle pane shows the DN A profile of the total cell population, while the right-hand pane consists of the DNA profile of BrdU-labeled cells only (gated on Rl).

Some plots of several bivariate distributions are provided in Figure 3.5 to Figure 3.7. In each case, the variance-covariance matrix, S, is given, followed by a scatter plot... 

FIGURE 4.3 Graphical illustration showing the effect of mean centering on a bivariate distribution of data points, (a) Original data, (b) Mean-centered data. The original x-y-axes are shown as dashed lines. 

Johnson, N. L. 1949. Bivariate distributions based on simple translation systems. 

Accepting normal bivariate distribution, Pearson s correlation coefficient, defined... 

Fig. 2. Bivariate distribution of V79 cells perturbed by hydroxyurea. Cells were pulse-labeled with 10 pM BUdR and allowed to progress through their cycle for 3 h. 0.3 mM hydroxyurea was added for 5 h. The profile above was obtained 2 h after removal of hydroxyurea.

A radius-diameter diagram is defined as a bivariate distribution of the - data set compounds in the space defined by the molecular radius and diameter it provides a summary of the similarities among the molecule chemical shapes in the topological or geometrical space. 

Exercise 3.6 Consider a bivariate distribution (M = 2) with two internal coordinates and 2, and let us construct a four-point quadrature approximation, resulting from univariate quadratures of order N = N2 = 2. Knowledge of the first, 2N = 2N2 = 4, pure moments with respect to the first, f, and second, 2, internal coordinates, suffices for... 

Most daughter distribution functions can be easily extended to bivariate problems. Let us consider two examples. In the first example particles with two components A and B are described. The particulate system is defined in terms of the size of these particles dp and the composition of the particles 0, expressed for example as the mass fraction of component A in the particle. When a particle breaks we can assume for example that the amount of component A is partitioned among the daughters proportionally to the mass of the fragments. Under these hypotheses, and the additional assumption of binary breakage following the beta distribution, the resulting bivariate distribution is... 

The extension of the DQMOM to bivariate and multivariate systems uses a quadrature approximation of order N. The simple case of a bivariate distribution uses... 

Figure 7.7. A bivariate distribution (a) as represented in a Monte Carlo simulation at the beginning of the simulation and (b) when a steady state is reached.

Figure 7.8. A comparison of the bivariate distribution, as represented in a Monte Carlo simulation, and the representation with the CQMOM with (a) Ai = 3, A2 = 1 and (b) N = 3, A2 = 2 at an intermediate instant of the simulation.

Fig. 2. Identification of apoptotic cells by LSC based on high values of maximal pixel detecting red fluorescence or fractional DNA content of propidium iodide (PI) stained cells. Exponentially growing human leukemic HL-60 cells, untreated (A) or induced to undergo apoptosis by treatment with camptothecin (B) (refs. 26,28), were stained with PI in the presence of RNase as described in the protocol. The scatterplots represent bivariate distributions of cells with respect to their integrated red fluorescence (proportional to DNA content) vs maximal red fluorescence pixel value. Only mitotic cells (M) have high maximal pixel value in the untreated culture. Apoptotic cells (Ap) that are present in the CPT treated cultures, are characterized either by the increased intensity of maximal pixel of red fluorescence or by a low tsub-Gj) DNA content. The relocation feature of LSC allows one to observe morphology of the cells selected from particular regions of the bivariate distributions. Upon the relocation, the cells with high maximal pixel value or with fractional DNA content show chromatin condensation and nuclear fragmentation, typical of apoptosis (panels on right).

Fig. 5. Detection of apoptotic cells based on the presence of DNA strand breaks. U937 cells were untreated (A) or treated TNF-a in the presence of cycloheximide (B) (refs. 26,28). The cells were then subjected to DNA strand break labeling and DNA staining as described in the protocol. The bivariate distributions (scatterplots) allow one to identify apoptotic cells as the cells with DNA strand breaks, and reveal the cell-cycle position of cells in either apoptotic or nonapoptotic population.

Sites cross-classified by number of accidents in both periods Figure 3.3 Bivariate distribution of accidents per site in Lothian region. 

---