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AMS 311 May 2, 2000 Two Problem Quiz next Thursday, May 4: univariate transformations, bivariate transformations. Quiz returned in exam week office hours. Final Examination: Thursday, May 11, 8:30-11:30, Old Chemistry 144. Exam week office hours: Professor Finch, Math 1-112: Friday, May 5: 2:30-4; Monday, May 8, 10-12; Tuesday, May 9, 10-12; Wednesday, May 10, 10-12. Mr. Wang, Math 3-129: Monday, May 8, 7-9 pm; Tuesday, May 9, 7-9 pm; Wednesday, May 10, 7-9 pm. Another problem using two variable transformations Let X and Y be independent, identically distributed random variables that are exponential with mean 1. Find the joint pdf of X-Y and X+Y. Conditioning on random variables. Recall that E ( X |Y ) (Y ) is a random variable (called the regression function). There are two fundamental identities: E ( E ( X |Y )) E ( X ). This is deceptively easily stated. Make sure that you understand the probability measure governing each expectation. Similarly, var( X |Y ) is also a random variable (it is also a function of Y). The second fundamental identity is var( X ) E (var( X |Y )) var( E ( X |Y )). The second fundamental identity is reflected in the basic identity of the statistical analysis of linear models: Total sum of squares=sum of squares due to model and sum of squares due to error. The most common example of correlated random variables is that of the bivariate normal distribution. Central Limit Theorems The most basic applications are concerned with sums and averages of random samples. Remember that a random sample of size n from the random variable X is defined to be a set of n independently and identically distributed random variables, each with the same marginal distribution as X . As you learned in AMS 310, the two basic random variables we are concerned with are n Sn n i 1 S X i and X n n n X i 1 n i . The basic moment calculations that we studied in the last chapter give you that E ( S n ) nE ( X ) and E ( X n ) E ( X ). The variance calculations give you that var( S n ) n var( X ) and var( X n ) var( X ) / n. The central limit theorem adds the fact that the distribution of these random variables becomes closer to normal as the number in the random sample increases. Modern proofs use a function called the moment generating function. For those who know complex analysis, there is a generalization of the moment generating function called the characteristic function that is preferred (because it always exists). The moment generating function of the random variable X is defined to be E(eXt), when it exists. The reason for the name is that the moments of a random variable can be obtained by success differentiation of the moment generating function. Example Problems Find the moment generating function for a Bernoulli trial with probability of success p. Find the moment generating function for a binomial random variable with n trials and probability of success p.