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# PROBABILITY THEORY 1 STA 6466

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Date Created: 09/18/15

STA 6466 Probability Theory 1 Fall 2007 Exam 2 Review Generalities o The exam will focus on Sections 22 through 46 of the notes 0 Study all the examples They help to know what is true and what is not and they may help you to construct your own counterexamples 0 Be familiar with the assigned homework problems 0 Do the suggested problems or as many as you can Chapter 2 Section 22 0 De nition of the integral p 54 you should know this 0 The basic algorithm for proving many results involving integrals is Step 1 Prove the result for indicator functions if necessary Step 2 Prove it for nonnegative simple functions usually use linearity of the integral here Step 3 Use approximation by simple functions Theorem 2113 and the Mono tone Convergence Theorem MCT to prove it for nonnegative func tions Step 4 Prove it for general functions using the fact that ffdlu ff 1 7 f f dw Concerning step 3 note that we actually proved a version of the MCT for a sequence of nonnegative simple functions increasing to a function 1 Theorem 222 so that we could use this mode of proof before we had proved the MCT In fact we used this result in proving the MCT itself pp 59760 Theorem 226 Obviously you should know the basic properties of the integral Part vi could just be replaced by the stronger result of Corollary 227 Don t worry too much about the proofs of this stuff 0 Section 223 The basic limit theorems for integrals are the Monotone Convergence Theorem MCT Fatou s Lemma and the Dominated Convergence Theorem DCT These are all very important and you need to be able to use them 7 Corollary 228 is a handy consequence of the MCT If you use it you can just say something like by a corollary of the MCT77 7 The extended MCT Corollary 229 can also be useful and it can be employed to modify the proof of Fatou s lemma to concoct an extended version of it as well Mainly though you should focus on the basic versions of these results 7 Remember when using the DCT that the dominating function is not allowed to depend on n For some reason that I don t understand this error is common Theorem 2215 is useful and its proof is a good illustration of the use of the DCT I m not too likely to ask about it on the exam but I could Theorem 2218 is important particularly part i which is used to show that Radon Nikodym derivatives are unique Theorem 2219 computing an integral using a density is used all the time Its proof is a simple application of the usual algorithm Theorem 2220 Change of Variables This is an important result and its proof is also a straightforward application of the usual algorithm Note that this theorem usually together with by Theorem 2219 justi es the way that we almost always compute expectations in practice Theorem 2221 and its corollaries The reason for writing these down is just to simplify certain calculations without having to think to hard about it They are not otherwise important You should know and understand the following basic facts about Riemann integrals and integrals with respect to Lebesgue measure 7 If the Riemann integral over a close and bounded interval exists then so does the Lebesgue integral and they are equal 7 The Lebesgue integral can exist when the Riemann integral does not you should know at least one example of this 7 lmproper Riemann integrals may exist when the corresponding Lebesgue integral does not exist 7 However when both the improper Riemann integral and the corresponding Lebesgue integral exist then it is practically a theorem that they are equal This never seems to be stated as a theorem but I suspect that this is just because there are too many different cases to consider In practice this can usually be proven by applying an appropriate limit theorem eg MCT or DCT to the Lebesgue integral 7 Anything more here Section 23 o Radon Nikodym Theorem Another very important result though obviously the proof is too long and technical to ask about 0 Lebesgue Decomposition you should know about this result 0 Jordan Decomposition I am not too likely to ask anything about this but note that it does tell you how to de ne the integral with respect to a signed measure De nition 236 Product Spaces and Fubini7s Theorem 0 Product Spaces you should be clueful about measurable rectangles and the de nition of the product sigma eld De nition 241 simple results like Proposition 244 and the de nition of product measure Theorem 245 but don t worry to much about the details of the construction 0 Fubini7s Theorem the two most important things are knowing how to check that it can be applied and being on the lookout for clever ways to apply it eg to get Ele foOOPle gt 75W You should know a bit about sections of sets Note that the proof of Fubini basically follows the usual algorithm it s just a lot of work to get started I m pretty unlikely to ask about any details of this proof 0 Theorem 2413 Integration by Parts The proof is instructive but I probably won t ask you to apply the result itself on the exam 0 Section 244 ln nite Products of Probability Spaces You should know what a cylinder set is the de nition of the in nite product a eld and the de nition of the in nite product probability measure which also speci es how to compute the probability of a cylinder set We will reVisit this brie y next semester running out of steam and time Chapter 3 You should pretty much know all the de nitions and theorems here and read through at least the shorter proofs a few times Chapter 4 The same general comment applies to this chapter The probability setting might be empha sized over the case of general measure spaces but you should be familiar with the general de nitions and have a pretty solid grasp of what is and isn t true in both settings

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