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Suppose that X and Y are discrete random variables. If the events X x and Y y are independent events for all x and y, then we say that X and Y are independent random variables. In such case, P(X or equivalently f (x, y) f1(x)f2(y) (28) x, Y y) P(X x)P(Y y) (27)
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Conversely, if for all x and y the joint probability function f(x, y) can be expressed as the product of a function of x alone and a function of y alone (which are then the marginal probability functions of X and Y), X and Y are independent. If, however, f (x, y) cannot be so expressed, then X and Y are dependent. If X and Y are continuous random variables, we say that they are independent random variables if the events X x and Y y are independent events for all x and y. In such case we can write P(X or equivalently F(x, y) F1(x)F2(y) (30) x, Y y) P(X x)P(Y y) (29)
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where F1(z) and F2(y) are the (marginal) distribution functions of X and Y, respectively. Conversely, X and Y are independent random variables if for all x and y, their joint distribution function F(x, y) can be expressed as a product of a function of x alone and a function of y alone (which are the marginal distributions of X and Y, respectively). If, however, F(x, y) cannot be so expressed, then X and Y are dependent. For continuous independent random variables, it is also true that the joint density function f(x, y) is the product of a function of x alone, f1(x), and a function of y alone, f2(y), and these are the (marginal) density functions of X and Y, respectively.
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Given the probability distributions of one or more random variables, we are often interested in finding distributions of other random variables that depend on them in some specified manner. Procedures for obtaining these distributions are presented in the following theorems for the case of discrete and continuous variables.
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CHAPTER 2 Random Variables and Probability Distributions
1. DISCRETE VARIABLES Theorem 2-1 Let X be a discrete random variable whose probability function is f(x). Suppose that a discrete random variable U is defined in terms of X by U (X), where to each value of X there corresponds one and only one value of U and conversely, so that X (U). Then the probability function for U is given by g(u) f [ (u)] (31)
Theorem 2-2 Let X and Y be discrete random variables having joint probability function f(x, y). Suppose that two discrete random variables U and V are defined in terms of X and Y by U 1(X, Y), V 2 (X, Y), where to each pair of values of X and Y there corresponds one and only one pair of values of U and V and conversely, so that X 1(U, V ), Y 2(U, V). Then the joint probability function of U and V is given by g(u, v) f [ 1(u, v),
2(u,
(32)
2. CONTINUOUS VARIABLES Theorem 2-3 Let X be a continuous random variable with probability density f(x). Let us define U (X) where X (U) as in Theorem 2-1. Then the probability density of U is given by g(u) where g(u)|du| or g(u) f (x) 2 dx 2 du f (x)|dx| f [c (u)] Z cr(u) Z (33) (34)
Theorem 2-4 Let X and Y be continuous random variables having joint density function f(x, y). Let us define U 1(X, Y ), V 2(X, Y ) where X 1(U, V ), Y 2(U, V ) as in Theorem 2-2. Then the joint density function of U and V is given by g(u, v) where f (x, y) 2 g(u, v)| du dv | '(x, y) 2 '(u, v) f (x, y)| dx dy | f [ c1 (u, v), c2(u, v)] Z J Z (35) (36)
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