Remark 6.2.1.
Consider random variables \(Y_1, Y_2, ..., Y_n\) and a function \(U(Y_1, Y_2, ..., Y_n)\text{,}\) denoted simply as \(U\text{.}\) Then three of the methods for finding the probability distribution of \(U\) are as follows:
1. The method of distribution functions: This method is typically used when the \(Y\)’s have continuous distributions. First, find the distribution function for \(U\text{,}\) \(F_U(u) = P(U \leq u)\text{,}\) by using the methods that we discussed in Chapter 5. To do so, we must find the region in the \(y_1, y_2, ..., y_n\) space for which \(U \leq u\) and then find \(P(U \leq u)\) by integrating \(f(y_1, y_2, ..., y_n)\) over this region. The density function for \(U\) is then obtained by differentiating the distribution function, \(F_U(u)\text{.}\) A detailed account of this procedure will be presented in Section 6.3.
2. The method of transformations: If we are given the density function of a random variable \(Y\text{,}\) the method of transformations results in a general expression for the density of \(U = h(Y)\) for an increasing or decreasing function \(h(y)\text{.}\) Then if \(Y_1\) and \(Y_2\) have a bivariate distribution, we can use the univariate result explained earlier to find the joint density of \(Y_1\) and \(U = h(Y_1, Y_2)\text{.}\) By intergrating over \(y_1\text{,}\) we find the marginal probability density function of \(U\text{,}\) which is our objective. This method will be illustrated in Section 6.4.
3. The method of moment-generating functions: This method is based on a uniqueness theorem,
[provisional cross-reference: thm-6-1], which states that, if two random variables have identical moment-generating functions, the two random variable possess the same probability distributions. To use this method, we must find the moment-generating function for \(U\) and compare it with the moment-generating functions for the common discrete and continuous random variables derived in Chapters 3 and 4. If it is identical to one of these moment-generating functions, the probability distribution of \(U\) can be identified because of the uniqueness theorem. Applications of the method of moment-generating functions will be presented in Section 6.5. Probability-generating functions can be employed in a way similar to the method of moment-generating functions.
