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If the points in the joint probability distribution of X and Y that receive positive probability tend to fall along a line of positive (or negative) slope, ρ XY is near +1 (or −1). If ρ XY equals +1 or −1, it can be shown that the points in the joint probability distribution that receive positive probability fall exactly along a straight ...
where P(t) is the transition matrix of jump t, i.e., P(t) is the matrix such that entry (i,j) contains the probability of the chain moving from state i to state j in t steps. As a corollary, it follows that to calculate the transition matrix of jump t , it is sufficient to raise the transition matrix of jump one to the power of t , that is
In probability theory, the Bapat–Beg theorem gives the joint probability distribution of order statistics of independent but not necessarily identically distributed random variables in terms of the cumulative distribution functions of the random variables. Ravindra Bapat and M.I. Beg published the theorem in 1989, [1] though they did not ...
This rule allows one to express a joint probability in terms of only conditional probabilities. [4] The rule is notably used in the context of discrete stochastic processes and in applications, e.g. the study of Bayesian networks, which describe a probability distribution in terms of conditional probabilities.
A probability metric D between two random variables X and Y may be defined, for example, as (,) = | | (,) where F(x, y) denotes the joint probability density function of the random variables X and Y.
One use for the probability integral transform in statistical data analysis is to provide the basis for testing whether a set of observations can reasonably be modelled as arising from a specified distribution. Specifically, the probability integral transform is applied to construct an equivalent set of values, and a test is then made of ...
A sequence of random variables that are i.i.d, conditional on some underlying distributional form, is exchangeable. This follows directly from the structure of the joint probability distribution generated by the i.i.d. form. Mixtures of exchangeable sequences (in particular, sequences of i.i.d. variables) are exchangeable.
This case arises frequently in statistics; for example, in the distribution of the vector of residuals in the ordinary least squares regression. The X i {\displaystyle X_{i}} are in general not independent; they can be seen as the result of applying the matrix A {\displaystyle {\boldsymbol {A}}} to a collection of independent Gaussian variables ...