STAT 512 -- EXAM 1 REVIEW SHEET I. Finding the Distribution of a Function of a r.v. II. The Method of cdf's A. Functions of a Single Y B. Finding the support of the transformed r.v. C. Functions of two r.v.'s 1. Sketching the support of (Y_1, Y_2) 2. Identifying the region where U <= u 3. Integrating the joint pdf over this region 4. Using appropriate limits of integration for double integral 5. Differentiating to get pdf of U III. The Method of Transformations A. Functions of a Single Y 1. Requirement of one-to-one function 2. Finding inverse function 3. Using transformation formula 4. How to Break Non-monotone function into regions of Monotonicity B. Probability Integral Transformation C. Bivariate Transformation Technique 1. Finding both inverse functions 2. Finding Jacobian of the transformation 3. Using Bivariate Transformation Formula 4. Identifying correct support for joint pdf of transformed r.v.'s 5. Integrating to get marginal pdf of one of the new r.v.'s 6. The use of an auxilary variable to create a bivariate transformation IV. The Method of Moment Generating Functions A. Uniqueness Property of mgf's B. Finding the mgf of U in terms of the mgf of Y C. Recognizing the form of the "common" mgf's D. Theorem about the mgf of a sum of independent r.v.'s E. Distribution of a linear combination of independent normal r.v.'s F. When does the method of mgf's work? V. Interesting Relationships between Distributions we have seen A. Standard exponential and Weibull B. Standard normal and chi-square C. Uniform and Cauchy D. Two indep. std. exponentials and uniform E. Two indep. std. uniforms and triangular F. Gamma and chi-square G. Indep. exponentials and gamma H. Indep. std. normals and chi-square VI. Order Statistics A. Definition of order statistics B. Deriving the distribution of the maximum C. Deriving the distribution of the minimum D. The distribution of the k-th order statistic E. The joint distribution of the j-th and k-th order statistics (j < k)