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Unformatted text preview: TRANSFORMATIONS OF RANDOM VARIABLES 1. I NTRODUCTION 1.1. Definition. We are often interested in the probability distributions or densities of functions of one or more random variables. Suppose we have a set of random variables, X 1 , X 2 , X 3 , . . . X n , with a known joint probability and/or density function. We may want to know the distribution of some function of these random variables Y = φ (X 1 , X 2 , X 3 , . . . X n ). Realized values of y will be related to realized values of the X’s as follows y = Φ ( x 1 , x 2 , x 3 , ··· , x n ) (1) A simple example might be a single random variable x with transformation y = Φ ( x ) = log ( x ) (2) 1.2. Techniques for finding the distribution of a transformation of random variables. 1.2.1. Distribution function technique. We find the region in x 1 , x 2 , x 3 , . . . x n space such that Φ (x 1 , x 2 , . . . x n ) ≤ φ . We can then find the probability that Φ (x 1 , x 2 , . . . x n ) ≤ φ , i.e., P[ Φ (x 1 , x 2 , . . . x n ) ≤ φ ] by integrating the density function f(x 1 , x 2 , . . . x n ) over this region. Of course, F Φ ( φ ) is just P[ Φ ≤ φ ]. Once we have F Φ ( φ ), we can find the density by integration. 1.2.2. Method of transformations (inverse mappings). Suppose we know the density function of x. Also suppose that the function y = Φ ( x ) is differentiable and monotonic for values within its range for which the density f(x) = 0. This means that we can solve the equation y = Φ ( x ) for x as a function of y. We can then use this inverse mapping to find the density function of y. We can do a similar thing when there is more than one variable X and then there is more than one mapping Φ . 1.2.3. Method of moment generating functions. There is a theorem (Casella [2, p. 65] ) stating that if two random variables have identical moment generating functions, then they possess the same probability distribution. The procedure is to find the moment generating function for Φ and then compare it to any and all known ones to see if there is a match. This is most commonly done to see if a distribution approaches the normal distribution as the sample size goes to infinity. The theorem is presented here for completeness. Theorem 1. Let F X (x) and F Y (y) be two cummulative distribution functions all of whose moments exist. Then a: If X and Y hae bounded support, then F X (u) and F Y (u) for all u if an donly if E X r = E Y r for all integers r = 0,1,2, . . . . b: If the moment generating functions exist and M X (t) = M Y (t) for all t in some neighborhood of 0, then F X (u) = F Y (u) for all u. For further discussion, see Billingsley [1, ch. 21-22] . Date : August 9, 2004. 1 2 TRANSFORMATIONS OF RANDOM VARIABLES 2. D ISTRIBUTION F UNCTION T ECHNIQUE 2.1. Procedure for using the Distribution Function Technique. As stated earlier, we find the re- gion in the x 1 , x 2 , x 3 , . . . x n space such that Φ (x 1 , x 2 , . . . x n ) ≤ φ . We can then find the probability that Φ...
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