Theories Joint Distribution
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ST 371 (VIII): Theory of Joint Distributions So far we have focused on probability distributions for single random vari- ables. However, we are often interested in probability statements concerning two or more random variables. The following examples are illustrative: • In ecological studies, counts, modeled as random variables, of several species are often made. One species is often the prey of another; clearly, the number of predators will be related to the number of prey. • The jo in t pr obabil it y di st ribution of the x, y and z compo nents of wind velocity can be experimentally measured in studies of atmospheric turbulence. • The joint distribution of the values of various physiological variables in a population of patients is often of interest in medical studies. • A model for the joint distribution of age and length in a population of fish can be used to estimate the age distribution from the length dis- tribution. The age distribution is relevant to the setting of reasonable harvesting policies. 1 Joint Distributi on The joint behavior of two random variables X and Y is determined by the joint cumulative distribution function (cdf ): (1.1) F XY (x, y ) = P (X ≤ x, Y ≤ y ), where X and Y are continuous or discr ete. For example, the probabilit y that (X, Y ) belongs to a given rectangle is P (x 1 ≤ X ≤ x 2 ,y 1 ≤ Y ≤ y 2 ) = F (x 2 , y 2 ) − F (x 2 , y 1 ) − F (x 1 ,y 2 ) + F (x 1 ,y 1 ). 1
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