Review of Elementary Probability Theory Math 495, Spring 2011

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1 Revie of Elementary Probability Theory Math 495, Spring Probability spaces Definition. A probability space is a triple ( HT,,P) here ( 3Ñ H is a non-empty setà Ð33Ñ T is a collection of subsets of Hfor hich H T, the complement A - œ H ÏA ={ = H : =  A} of any member A of T is also in T, and, for each countable index set M and each choice of sets A 3ß 3 Mß in T, the union A= " 3 M A 3 is in T; Ð333Ñ P: Hp[0,1] is a function for hich P( H) =1 and, for each countable collection A 3ß 3 M, of mutually disjoint members of T, P( " A ) œ! P(A Ñ Ð"Ñ 3 M M!Þ"Þ#Þ Terminology. When ( HT,,P) is a probability space, names for Hinclude the population space and the sample space. The collection T is called either the class of measurable subsets of H or the class of admissible outcomes or the class of events. The function P on T is called the probability measure (shortened version of "function that measures probability for events").!þ"þ$. Examples. ( 3Ñ H is a countable set, T is the collection of all subsets of H, and P(A) œ! : here = È : A is a function from Hinto [0,1] for hich = A =! : œ "Þ We then say that ( HT,,P) is a discrete probability space and = H = usually limit attention to the case hen each number : = is >0. For many applications of probability theory, discrete probability spaces are all that's needed. Most courses in probability begin ith the very special case hen H is a finite set ith N elements and each : = is equal to 1/N. This is fine for computing various probabilities for card and dice games but otherise it's necessary to move on to countably infinite sets H and nonconstant p = 's hose sum is 1. Even for something as easy as probabilities for a flipped coin to come up heads or tails, H has to be some horribly abstract countably infinite set consisting of, in principle, "all" possible flips of a certain coin ith various specifications on flips (air conditions, surface on hich the

2 coin lands, manner in hich the coin is flipped, alloed height for a flip,...). The to events of interest are "heads" and "tails" and these are disjoint subsets of Hhose union is equal to Hprovided e disallo flips here the coin lands on edge and doesn't fall over on one side or the other. ( 33Ñ For some natural number N, H is the N-dimensional R R R rectangle [0,1] in ß T= U is the smallest class of subsets of H hich is closed under countable unions and contains all open or closed subrectangles of H(members of U R are called Borel setsñ, and P(A) ÐA) is the R # dimensional volume of A. When A is a finite union of rectangles, P(A) is defined as in multi-variable calculus courses. Math 4021 proves that the N # dimensional volume function extends from finite unions of rectangles to general Borel sets and satisfies the countable union property (1). Often, "true" applications of probability theory involve a very complicated countably infinite space H hich e can "approximate" by a dense collection of points in a large rectangle in N and go on to rescale the large rectangle by [0,1] R Þ The point is that if a subset A of our space H is ell approximated by a union of rectangles, it's much easier to add up the volumes of these rectangles than compute the value of the infinite series! : = E = Þ ( 333Ñ In a course in measure theory such as Math 4122, it's shon that hen e have N probability spaces ( H 3 ßT 3 ßP 3 Ñß " Ÿ 3 Ÿ Rß e can construct a product probability space ( HT,,P) here H= H H â H is the Cartesian product of the H 's, P is the product " # R 3 3 R " # # R 3 3 3œ" 3 H3ß T of the P = in the sense that P(A A â A Ñ œ P ÐA Ñ for each choice of sets A in and is the smallest collection of subsets of Hhich contains all of these Cartesian products of members of the H 3 's and satisfies the properties in Definition This is very easy hen each of the factor spaces is discrete and is tantamount to redoing the multivariable calculus theory of R # dimensional volumes for continuous factor spaces as in ( 33ÑÞ With considerably more ork, one can go on to construct product spaces ith a countably infinite number of factors--for this, one has to get into the theory of infinite products in order to make sense out of the measure P on the infinite product space being the infinite product of the measures P. The details are fussy but the basis idea is the same as for finite 3 products. One might think that such infinite products are totally useless arcane artifacts. In fact, they are hidden in the construction

3 of i.i.d. sequences of random variables (see belo) and such sequences are mentioned in every elementary probability textbook and used heavily in probability theory Definition of Conditional Probabilities. When ( HT,,P) is a probability space and B is a member of T for hich PaB b>0, the conditional probability measure P B is defined on T by P(A$ B) P B(A) = P(B). (2) It's customary to rite P(A B) for P B(A) and to call it the conditional probability of A given B. It's very easy to check that the fact that P has the properties in 0.1.1( 333Ñ implies that P B has these properties as ell. When P B (A) =P(A) (equivalently, hen P(A $ B) =P(A)P(B)), A and B are said to be independent admissible outcomes. In calculations, e often rite P(A and B) for P(A $ B) =P(B) P (A) =(probablity of B) (probability of A given B). B 0.2 Random Variables on a Probability Space and their distributions Definitions of Types of Random Variables. Let ( HT,,P) be a probability space. ( 3Ñ A discrete random variable on His a function X from H into a countable set S for hich the event X= Bß 3Þ/Þß the set H B œ { = H: X( = )= B ß lies in T for each B in S. Then 0X( BÑ œ P(X= BÑ defines the distribution of X. (other names for 0 X occur in old probability textbooks but these alternative names are no out of vogue). Vice versa, e say a distribution on S is a function 1:S p[0,1] for hich! 1ÐBÑ œ "Þ As in Example 0.1.3( 3Ñß e could extend 1 to be a probability measure on the class T(S) consisting of all subsets of S. Alternatively, as long as e can carve up H into the countable union of mutually disjoint subsets H B ith P( HBÑ œ1( BÑßthere is a random variable X on Hfor hich 1= 0 X. [When our original space doesn't decompose in this ay, e can alays cook up another probability space for hich 1 is the distribution of a random variable on the ne space]. When our countable set S happens to have a "natural" description by some countable subset of R ßR 1ß so be it. Otherise, e can alays choose a ay to code the members of S by a standard countable index set such as I8 œ Ö!ß "ß âß 8 # " hen S is finite ith 8 elements and by & œ! " Ö! œ Ö!ß "ß #ß â hen S is countably infinite. B W

4 Usually, S is then just identified ith the coding index set. Codings are handy but unnecessary except in the cases here there's only one "natural" ay to code S and e ant to go on to do "meaningful" arithmetic calculations ith codes.. ( 33Ñ A scalar-valued random variable on H (or ordinary random #" variable) is a function X: H p for hich X ÐB) ={ = H : X( = ) B} is a member of T for each choice of B in U " Ðsee 0.1.3( 33). We customarily describe X #" ÐB) by {X B} or "the event that X lies in B" and obtain a " probability measure P\ on P\ ÐFÑ œ T ÐX lies in B). It turns out that a necessary and sufficient condion for {X B} to be in T for each Borel set B is that this property holds for all semiinfinite intervals B B œð# _ß B] and this is the motivation for defining the cumulative distribution function Ð-Þ.Þ0Ñ F\ of X as the function from into [0,1] given by F\ ÐBÑ œ P(BBÑ œ P(X Ÿ BÑ. It follos easily that P\ Ð+ß b]=p( + ' BŸ,Ñis equal to F\ Ð,Ñ# F \( +Ñ for all +ß, in ith + ',. Not so easy is the fact that F\ uniquely determines the measure P\ on U " Þ In various probability books, one can find conflicting definitions of " the statement that X is a "continuous random variable. This alays means in particular that X( H Ñis a non-countable subset of ß F\ is continuous, and there is a function 0\ À pò!ß_ñfor hich F \ ÐBÑ œ ' B #_ 0\ Ð>Ñ.>Þ But this anti-derivative stipulation is much stronger than just saying that F\ is continuous. Some authors stipulate that 0 \ should be continuous except for countably many jump discontinuities (elementary texts only allo finitely many discontinuities), others eaken this to 0 \ being a Riemann-integrable function, hile authors of sophisticated texts eaken things further to 0 \ being only Lebesgue measurable (the technical definition of this is that 0 #" Ð# _ß BÓ is in U " for each B and this allos the possibility that there is X no point here 0 \ is continuous). They then go on to prove a theory in measure theory saying that 0 \ exists if and only if F \ satisfies a technical property called absolute continuity (this property is too complicated to spell out here). When it exists, 0 \ is called the probability density function (:Þ.Þ0ÞÑ for the continuous <Þ@Þ X ( <Þ@Þ is a common shorthand for random variable). As in ( 3Ñß all of this is reversible in the sense that hen 0À pò!ß_ñis any Lebesgue measurable function for hich ' _ #_ 0ÐBÑ.B œ "ß then one can cook up a probability space and a continuous <Þ@Þ X on this space for hich 0 œ 0\ Þ Despite all of this hard theoretical machinery, suffice it to say that there are very good reasons hy essentially

5 all probabilists limit their examples of non-discrete = to the continuous = ith :Þ.Þ0Þ = having at most finitely many jump discontinuities ## nothing less than this gives nice "orking models" for practical applications. ( 333Ñ A vector-valued random variable on H is just an 8 # >?:6/ X= ÐX" ßX # ßâßX8Ñ of scalar-valued <Þ@Þ = on H. We then just "turn the crank" on everything in ( 33Ñ, replacing systematically ith 8 Þ Thus, X #" ÐB) T for each B U 8 ßT\ ÐB) =P(X #" ÐB))=P(X lies in B) defines a probability measure on U 8, F\ ÐB" ßB# ßâßB8Ñ œtð\ 3 ŸB 3for "Ÿ3Ÿ8Ñ defines the joint c.d.f. of ÐX" ßX# ßâßX8Ñß and \ is said to be continuous if there is a Lebesgue measurable function 0\ À 8 pò!ß_ñ for hich P X(B) œ' B0\ ÐB" ßB# ßâßB8 Ñ.B ".B# â.b8 Ð3) for every Borel subset B in 8 Þ When it exists, 0\ is called the joint p.d.f of X= ÐX" ßX # ßâßX8ÑÞ The fact that absolute continuity of F\ is necessary and sufficient for the existence of 0\ and that absolute continuity is very hard to check is by-the-by. Non-discrete vector-valued <Þ@Þ = are just blithely assumed to have nice joint :Þ.Þ0.'s in all practical modeling applications Independent Random Variables and I.I.D. Sequences ( 3Ñ Definition. When ( HT,,P) is a probability space and X, Y are random variables on H (usually either both discrete ith values in the same countable set S or both scalar-valued), e say X and Y are independent if P(X is in A and Y is in B)=P(X is in A)P(X is in B) for all possible events A for X and B for Y. Whether or not X and Y are independent, they are identically distributed if both are continous <Þ@ ' = ith the same :Þ.Þ0Þ or both are discrete ith the same distribution functions as in 0.2.1( 3Ñ. ( 33Ñ Definition. A sequence (X 8Ñ 8! of <Þ@Þ = on the same probability space is independent and identically distributed Ðfor short, an 3Þ3Þ.Þ =/;?/8-/Ñ if each pair of members of the sequence are both independent and identically distributed. Ð333ÑÞ Remarks. It's easy to build finite sequences X" ß X# ß âx8of mutually independent and identically distributed <Þ@Þ's. One just starts ith a fixed <Þ@Þ X on some probability space ( HT,,P) and looks at the

6 8 Ð8Ñ Ð8Ñ Cartesian product space ( H ß T ß P Ñ as in Example 0.1.3( 333Ñ ith the X3 = defined on this product space (sometimes called the space of samples of size 8Ñ by X3 Ð= " ß = # ß âß = 8Ñ œ Xa= 3bÞ The 3Þ3Þ.Þ property is then very easily checked. This construction is behind a large part of the arsenal of statistics (applied probability). But, as mentioned before, it's not so easy to generalize this finite product construction to infinite products (at least, not so easy for those not comfortable ith measure theory) and its seems more than a little bit of a disservice to readers of elementary probability books that the authors of these books just define infinite 3Þ3Þ.Þ sequences, pretend that they're little different from finite 3Þ3Þ.Þ sequencesß proceed to rattle on at length over theoretical properties of 3Þ3Þ.Þ sequences, and never give an example of an infinite 3Þ3Þ. sequence for the good reason that infinite products are the only ay in hich they arise. 0.3 Stochastic Processes Definition. A stochastic process is a collection ax > b> X of random variables X> each of hich is defined on some fixed probability space ( HT,, P) and has values in some set S. S is called the state space and its members are called states of the process. T is the collection of times for the process. S can be either a countable set (hich can then be coded as in 0.2.1( 3ÑÑ or a Borel subset of, /Þ1Þ an interval in. The discrete time processes are those for hich T= & hile continuous time processes take T to be an interval in, usually a closed interval ith distinct endpoints such as [0,b] or [0,_). When S is countable, all of the random variables X> are discrete and have distribution functions as in 0.2.1( 3Ñ; to get anyhere, assumptions have to be made about joint distributions for these random variables. When S isn't countable, it's customary to assume each X> is a continuous <Þ@Þ and go on to make assumptions about :Þ.Þ0Þ = for the individual random variables and 8 # >?:6/= of these random variables (joint :Þ.Þ0Þ =Ñ. In all cases, the manner in hich the process evolves over time depends on the assumptions made about distributions Definition. A Markov process is a stochastic process here, roughly speaking, at each "current time" >ß! the behaviour of the process for future times > ) >! is controlled by its behviour at >! ithout regard to the history of the process at past times > ' >Þ! At least intuitively, eather phenomena (temperature, precipitation, />-ÞÑ in each locality are Markov processes but so complicated that future predictions (conditional

7 probabilities) based on current conditions are extremely complicated and virtually impossible for times ell into the future (/Þ1Þß six months from no). There are many other biological, physical, business, and economic phenomena hich can be modeled as Markov processes. For this reason, Markov processes are a big topic in probability theory. Not surprising, the Markov processes here one can prove nice theorems and explicitly do calculations tend to be oversimplified versions of real orld processes. 0.4 Notations in Probability In most parts of mathematics, e use generic symbols for various objects (functions 0ß sets S, matrices A, linear operators T, />-ÞÑ but then feel obliged to use subscripts, superscripts, and other notational devices or to use closely related letters hen e use an object to construct similar objects. In this vein, subscripts and superscripts ere used above to pass from an initial probabilty measure P on T to various other measures Ð8Ñ Ð_Ñ PFß PBß P ß P hich e construct from P and hich, aside from PFß act on different classes of subsets in different probability spaces. This is not the common practice in probability books and research articles, here, to the maximum extent possible, every probability measure is denoted by P and decorating subscripts, superscripts, â don't appear. This likely arises from the early days of the subject (before it as recognized that probability theory is part of measure theory) hen P(E) started to be used as a general shorthand for the "probability/odds that the event E ill occur in the future". In the early days, nothing much as said about the types of events to hich it makes sense to assign a probability. Hence, the modern vie that each probability measure is a function on a certain class T of subsets of a set H ouldn't have made any sense to early probabilists. As ith most subjects, the early history strongly influences notations for a very long time. Briefly, readers of probability books and articles just have to get adjusted to the fact that, unless confusion is inevitable, every type of function hose values are in [0,1] is most likely denoted by P, even if many different types of such functions are simultaneously entering into the discourse. One can, as many have done in the past, get angry and complain loudly about the difficulties this causes in trying to comprehend hat's being said. But probabilists haven't been sayed by such harangues in the past and likely on't be sayed in the foreseeable future!

8 Despite the above lament, one can use common sense over the fact that 0 X is used generically both for the probability distribution of a discrete <Þ@ (or joint distribution for an 8 # tuple of discrete <Þ@Þ = ) and the :Þ.Þ0Þ of a continuous <Þ@Þ Ð or joint :Þ.Þ0Þ for an 8 # >?:6/ of continous <Þ@Þ =Ñ. In a context hen only discrete <Þ@Þ = or only continuous <Þ@Þ = are under discussion, there's no problem. In the rare cases, here both types of <Þ@Þ = are being simultaneously discussed, use different symbols, /Þ1Þsave 0 \ as a :Þ.Þ0Þsymbol and use something like 1\ for distribution functions of discrete <Þ@Þ =Þ Similarly, hen there are just too many different types of P's under discussion, don't be afraid to go "against the tide" by inventing your on decorating scheme. Note that -Þ.Þ0Þ = F\ make sense for every <Þ@Þ X ith values in or 8 ##the same definition applies hether X is discrete, continuous, or neither. But, there are also certain continuous random variables of so-called F-type used heavily in regression theory. Exactly the same comments as above apply--use common sense to avoid confusion.