Within a Bayesian modeling approach, the beta prior distribution for parameter [p.sub.i] is a conjugated prior when the data are

binomially distributed given the response rate.

Defaults are distributed

binomially and so most of the probability mass for the default distribution is located within two standard deviations [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the expected value of the distribution [a portfolio loss rate of LGD x PD].

Add to this the fly-tipping of large, inorganic waste such as refrigerators, mattresses and building waste, often discarded into inaccessible gorges, and the problem expands

binomially.

Seed MC and shoot height were

binomially related (R2=0.88) with height between 6.93 and 5.18% MC all significantly higher than the control.

Conditional on [[subset].sub.h [is an element of] W] [V.sub.h] = [v.sub.h], [V.sub.a] and [V.sub.b] are distributed

binomially, with parameters [[Pi]*.sub.a] = [[Pi].sub.a]/([[Pi].sub.a] + [[Pi].sub.b]), [[Pi]*.sub.b] = [[Pi].sub.b]/([[Pi].sub.b] + [[Pi].sub.a]), and m* = m - [[Sigma].sub.h [is an element of] W] [v.sub.h].

To account for the fact that the responses are

binomially distributed, we model the variance between responses as [z.sub.ij] [e.sub.ij], where

Ordinary least squares (OLS) analysis is inappropriate when the dependent variable is binary, because OLS assumes a linear additive model with normally distributed error terms, while the true probability model is nonlinear with

binomially distributed errors (Aldrich and Nelson, 1984).

Green's method compares the variance (or in this case its estimate, the mean squared error [MSE]) of the number of males observed with that expected if the number of males were

binomially distributed.

where [[Xi].sub.t] denotes a random shock whose outcome is

binomially distributed being realized as:

Under the null hypothesis that scores are attributed randomly and independently, this results in a

binomially distributed count (number of high responses out of 6) as the response variable, and logistic regression (McCullagh & Nelder, 1985, chapter 4) can then be used to analyze the responses.

Rather, the equations are each converted into a logit form by solving for the number of council seats held by Hispanics, which is assumed to be distributed

binomially. Estimates of ln([beta]), [gamma], and [delta] are then obtained by maximum likelihood estimation.

If individuals move from one size class to the next with a constant probability, then the frequency distribution of the number of individuals changing size class will be a

binomially distributed random variable.