Outline of Elementary Probability Theory Math 495, Spring 2013

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1 Outline of Elementary Probability Theory Math 495, Spring Þ Probability spaces 1.1 Definition. A probability space is a triple ( H, T,P) here ( 3ÑH is a non-empty setà Ð33ÑT is a collection of subsets of H for hich H T, the complement A - œ H ÏA ={ = H : =  A} in H of any member A of T is also in T, and, for each countable index set M and each choice of sets A3ß 3 Mß in T, the union A= 3 MA 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 ÑÞ Ð"Ñ 3 M M Because the complement of a countable union is the interection of the complements, Ð33Ñ implies that T is closed under countable intersections - and, in particular, is closed under set differences A\B =A B œ Ö= A: =  B}. Among the easy consequences of (1) are that P(A) Ÿ P(B) hen A and B are in T ith A B; indeed, since B = A (B\A), e have P(B)=P(A) + P(B\A) ith P(B\A) 0. For the special case B= H, it follos that P(A - Ñ œ 1 P(A). 1. #. Monotonicity properties of P. Suppose ( H, T, P) is a probability space. ( 3) When (A3Ñ3 is a collection of members of T indexed by the set ={0, 1, 2, â} of natural numbers hich is monotone increasing in the sense that A3 A3 " for each 3, then A= - 3 is the disjoint union of A and the sets A ÏA and it follos from (1) that P(A) = P(A Ñ " 3 3 " " _ 8! ÐP(A Ñ P(A ÑÑ œ P(A Ñ lim! ÐP(A Ñ P(A ÑÑ 3 3 " " 3 3 " 3œ# 8Ä_ 3œ# œ lim 8Ä_ P(A 8 ÑÞ ( 33) When (B 3 Ñ 3 is a collection of members of T indexed by hich is monotone decreasing in the sense that B3 ª B 3 " for each 3 ß then putting A3 œ HÏB3ß e have A3 A 3 ", P(B3Ñ œ " P(A3Ñß and B= + B - - œ Ð A Ñ. It follos from ( 3) that P(B)= lim P(B ÑÞ Ä_

2 "Þ$Þ Terminology. (3Ñ When ( HT,,P) is a probability space, names for H include the population space and the sample space. The collection T is called either 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") and, for each event A, P(A) is called the probability of A. When P(A) =0, e say A is a null event.. Ð33ÑIn general, any collection of subsets of a non-empty set H satisfying the conditions in 1.1( 33) is said to be a 5-algebra of subsets of H. If f is any non-empty collection of subsets of H, it's easy to check that the intersection T( f) of the family of 5-algebras containing f is a 5 algebra contained in every 5-algebra hich contains f or, more simply, the smallest 5-algebra containing f. T( f) is called the 5-algebra generated byfþ The 8 5 -algebra of subsets of generated by the collection of half-open 8-8 dimensional rectangles # Ð+ ß, ] ={ B œ ÐB ßâßB l+ B Ÿ, for each 3 3 " 8Ñ œ" is denoted by U and is called the algebra of Borel subsets of Þ It's not hard to see that U 8 is also generated by either the collection of all closed subsets of 8 or the collection of all open subsets. ( 333ÑA measure on a 5 algebra T of subsets of some set H is a function.: Tp [0,_) for hich (1) holds and e then say ( H, T,.) is a measure space. The nicest measures are those hich are 5-finite in the 8œ_ sense that there is a sequence (K8Ñ8 " in T for hich H= - 8œ" K 8 and. ÐA).(K Ñ _ for each 8Þ Then, ith T œ ÖA T:A K ß P ÐA)= (K 8 Ñ defines a probability measure on T8Þ In this ay, the study of 5-finite measures reduces to the study of probability spaces; vice versa, every probability space is also a measure space and all of the "deep" theorems about probability spaces use tools from measure theory. Math 4121 and Math are courses in measure and integration theory. We ill just quote the results e need from such courses in 495. Hoever, those ho ant to become experts in stochastic processes should ultimately take a course or to in measure and integration theory. T ( 3@Ñ We can alays complete a measure space ( H, T,.) by letting be the collection of sets of the form A=A N here A T and N is a

3 subset of some B T ith.(b)=0 and by defining.(a) to be.(a). It's easy to check that ( H, T,.Ñ is a measure space. Intuitively, completion just means e enlarge T by tossing in subsets of a null set and declaring these subsets to be null sets as ell. This turns out to be convenient in proving theorems about measure spaces. "Þ4. Examples of Probability Spaces ( 3Ñ We say that ( H, T,P) is a discrete probability space if H is a non- H empty set, T is the collection 2 consisting of all subsets of H, H and, for each A 2 ß P(A) œ!: here = È : A is a function from H = A = into [0,1] for hich!: œ "Þ This condition dictates that only countably = = H : = many of the numbers are >0 and it's customary to limit attention to the case hen H is countable and each : = =P({ = }) is >0 (otherise e could simply discard all of the elementary null events { = } here : = =0). For many applications of probability theory, discrete probability spaces are all that's neededà in modern physics, the physical universe can be modeled as a countable set and it follos that all "real orld" probability spaces are discrete. Most elementary probability courses 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 fair card and dice games but otherise it's necessary to move on to countably infinite sets H and nonconstant : = '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 ugly infinite set consisting of "all" possible flips of a certain coin ith various specifications on flips (air conditions, eight density for the coin, properties of the surface on hich the 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 H hose union is equal to H provided e disallo flips here the coin lands on edge and doesn't fall over on one side or the other. ( 33Ñ Lebesgue shoed in the early 1900s that the 8 dimensional volume function has a unique extension to a measure on U and it's customary to refer to -8 as 8 dimensional Lebesgue measure (Math 4121 and 5411 give the proof of the existence of - 8 ÑÞ The

4 8 8 8 class _ of sets obtained by completing Ð ß U ß-8Ñ is called the collection of Lebesgue measurable sets; it's customary to continue to denote the 8 8 extension of - 8 to _ by -8. Then, for each K U ith -8ÐK)=1, (K, 8 U(K)={B U À B K}, restriction of -8 to U(K)) is a probability space Ñ. Often, "true" applications of probability theory involve a very complicated countably infinite space H hich e can "approximate" by a dense collection of equally probable points in a large rectangle K in N hich, by rescaling, can be assumed to have volume 1. We then pass from the discrete probability space H to the "continuous" space K and use -8 on U(K) to estimate probabilities. ( 333Ñ A function F: p[0,1] is said to be a cumulative probability distribution function (for short, a -Þ.Þ0 ÞÑ if F is monotone increasing, F is continuous from the right in the sense that, for eachb ß F( BÑ is the limit of the terms F( CÑ as C > B approaches Bß lim FÐBÑ œ!ß and BÄ _ lim F( BÑ œ "Þ Using the tools established in measure theory courses, there BÄ_ is then a unique probability measure. F on U " for hich. F((a,b] =F(b) F(a) for each pair of real numbers a, b ith a<b. Vice versa, if P is any probability measure on U " ß the 1.2 monotonicity properties of P imply that FaBb œ P(( _ßBÓÑ defines a -Þ.Þ0Þ ith P =. F. In a similar ay, one can define -Þ.Þ0Þ = on 8 and use them to describe all probability measures on U 8 Þ We say 0 À pò!ß_ñ is a probability density function (for short, a :Þ.Þ0 ÞÑ if 0 is integrable (in either the Riemann or Lebesgue sense) ith ' _ -_ 0Ð>Ñ.> œ "Þ In this case, e have a -Þ.Þ0Þ F defined by B F( BÑ œ ' _ 0Ð>Ñ.>, F is everyhere continuous, and, except on a. F-null set, F is differentiable ith 0ÐBÑ œ F ÐBÑÞ Not every continuous -Þ.Þ0Þ arises in this ay from a :Þ.Þ0 Þ but those that don't are of little practical value. ( 3@Ñ In a course in measure theory, it's shon that hen e have N probability spaces ( H 3 ßT 3 ß P 3 Ñß " Ÿ 3 Ÿ Rß e can construct a product probability space ( HT,,P) here H= H" H# â HR is the Cartesian product of the H3's, T is the 5- algebra generated by the collection of Cartesian products A=A A â A ith A T for 1 Ÿ 3 Ÿ 8ß and P is the unique " # R 3 3

5 R T " # R 3 3 3œ" measure on for hich P(A A â A Ñ œ # P ÐA Ñ hen each A3 T3Þ It's customary to call P the product of the measures P3ß" Ÿ 3 Ÿ 8Þ One can similarly define products of finitely many measures Þ In particular, -8 is the product of 8 copies of the measure -" since e define 8 dimensional volumes of rectangles to be the products of the lengths of the sides. With considerably more ork, one can go on to construct product probability spaces ith an 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 infinite Cartesian products being the infinite product of the measures P on the factor sets. The details are fussy but the basis idea is the same as 3 for finite products. One might think that such infinite products are totally useless arcane artifacts. In fact, they are hidden in the construction of i.i.d. sequences of random variables (see belo) and such sequences used heavily in probability theory. Infinite product spaces are also used to construct Markov chains ith specified properties and more general stochastic processes. (@Ñ 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 1.1( 333Ñ implies that P B has these properties as ell. Then, ith TB={A T:A B}, (B, TBß P B) is a probability space. When P B(A) =P(A) (equivalently, hen P(A B) =P(A)P(B)), A and B are said to be independent events. 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 2 Random Variables on a Probability Space and their distributions 2.1 Definitions of Types of Random Variables. Let ( HT,,P) be a probability space. ( 3Ñ A discrete random variable on H is a function X from H into a " countable set S for hich the event X= Bß 3Þ/Þß the set H B œ X ÐBÑ { = H : X( = )= B ß lies in T for each B in S. Then 1 ( BÑ œ P(X= BÑ defines X

6 the distribution of X. (other names and symbols for 1 X occur in old probability textbooks but these alternatives are no out of vogue). Vice versa, e say a distribution on a countable set S is a function 1:S p[0,1] for hich! 1ÐBÑ œ "Þ As in Example 1.3( 3Ñß e could use 1 B W to define a probability measure Q on the class 2 consisting of all subsets of S; ith X( BÑ œ B, X is a discrete random variable on the discrete measure S space (S,2,Q) ith 1= 1X. Alternatively, if ( HT,,P) is a any measure space for hich H is the countable union of mutually disjoint subsets H B, B S, " ith P( HBÑ œ 1( BÑ for each B (/Þ1ÞH œ Ð!ß"Óß T œ _ Ð0,1], P is the restriction of -" to T, and the HB's are disjoint half-open intervals of length 1( BÑÑß there is a unique random variable X on H for hich 1= 1X and " H B œ X ÐBÑ for each B S. In this ay, there is "essentially" no difference beteen distribution functions and discrete random variables. When our countable set S happens to have a "natural" parametrization by some countable subset of R ß R 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"ßâß8 " hen S is finite ith 8 elements and by œ Ö! œ Ö!ß"ß#ßâ hen S is countably infinite. Usually, S is then just identified ith the coding index set. Codings are handy for calculations 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 is T measurable in the sense that X " ÐB) ={ = H : X( = ) B} is a member of T for each choice of B in U " Ðsee 0.1.3( 33). We customarily denote X " ÐB) by {X B} and call it " "the event that X lies in B"; e obtain a probability measure P\ on P\ÐFÑ œ PÐX 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 œ Ð _ßB] 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 PXÐ+ß b]=p( + X Ÿ,Ñ is equal to F\ Ð,Ñ F \( +Ñ for all +ß, in ith +,. From the general results mentioned above, F\ uniquely determines " " the probability measure P X on U and, for each B U ß P X(B)=P(X B) =P(X " ÐB)). S

7 In various probability books, one can find conflicting definitions of " the statement that X is a "continuous random variable. This alays means that X( H Ñ is a non-countable subset of ß F X is continuous, and there is a :Þ.Þ0Þ 0X À pò!ß_ñ for hich F \ ÐBÑ œ ' B _ 0\ Ð>Ñ.>Þ This antiderivative 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 _ßBÓ is in U " for each B and this allos the possibility that there is X no point here 0 \ is continuous). Using the last definition, there is a theorem 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.Þ0 ÞÑ 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 0 À pò!ß_ñ is 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 all probabilists limit their examples of non-discrete <Þ@Þ = to the continuous <Þ@Þ = 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" ß X# ß âß X8Ñ 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 ß T\ ÐB) =P(X " ÐB))=P(X lies in B) defines a probability measure on U 8, F\ ÐB" ßB# ßâßB8Ñ œ TÐ\ 3 Ÿ B 3 for " Ÿ 3 Ÿ 8Ñ defines the joint c.d.f. of ÐX" ß X# ßâß X8Ñß 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 ' B 0 \ ÐB " ßB # ßâßB 8 Ñ.B ".B # â.b 8= ' B 0 X. -8 Ð 3) for every Borel subset B in 8 Þ When it exists, 0\ is called the joint p.d.f of X= ÐX" ß X# ßâß X8ÑÞ 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

8 = are just blithely assumed to have nice joint :Þ.Þ0.'s in all practical modeling applications. 2.2 Independent Random Variables and I.I.D. Sequences ( 3Ñ Definition. When ( H, T,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 Borel sets A and B. Whether or not X and Y are independent, they are identically distributed if both are continous <Þ@ ' = ith the same :Þ.Þ0 Þ or both are discrete ith the same distribution function ( 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. As mentioned above, construction of 3Þ3Þ.Þ sequences is most easily accomplished by using countably infinite products of copies of a fixed probability space. Although 3Þ3Þ.Þ sequences are mentioned in elementary probability and statistics courses, students are often given the false impression that they are easy to construct. 2.3 Expected Values, Variances, and Co-variances for random variables. Ð3Ñ General Facts. In measure theory courses, it's shon that that for any measure space ( HT,,.) and any T-measurable function 0 À H p, one can define 0ll œ l0l.. as the limit of the Lebesgue # # N " ' H sums! " " 4. ÐX Ð4ß4 ÓÑ as N p_ and hen the limit is finite, 0 is 4œ" # R declared to be integrable ith respect to. ith ' H 0.. defined to be ' ' H7+BÐ0ß!Ñ.. H7+BÐ 0ß!Ñ.. Þ Note that, for each = Hß 0Ð= Ñ œ 7+BÐ0Ð= Ñß!Ñ 7+BÐ 0Ð= Ñß!Ñ ith 0Ð= Ñl being the sum of these to maxima. Hence, e alays have ' ' H0.. Ÿ H 0... " For the special case ( HT,,.) œ (,_ ß -" Ñß every Riemann integrable function 0 on is also integrable ith respect to - and then ' " 0. -" is equal to the Riemann integral ' 0ÐBÑ.B.

9 When T and. are fixed, it's customary to denote by L " ÐHÑ the vector space consisting of all integrable functions and go on to define L : ÐH) ß : "ß to be the vector space of all T-measurable functions : 0 À H p for hich l0l is integrable ith 0ll defined to be (' : : H l0l.. Ñ "Î: Þ Very important is the Riesz-Fischer Theorem asserting that : : L ÐHÑ is complete in the sense that, for each sequence ( 08Ñ8 " in L ÐHÑ for hich ll: p! as 8ß7p_ß there exists 0 L : ÐHÑ such that 0 08 ll: p! as 8p_ and there is a.-null set N for hich 0Ð= Ñ œ lim0 Ð= Ñ =  8Ä_ 8 for each N. Ð33Ñ Probability Definitions and Notations. When ( H, T, P) is a probability space and X L " ÐH Ñß ' H X. P is defined to be the mean or expected value of X and is denoted either by. X or by E(X). By the properties of Lebesgue sums, it follos that, hen X is a discrete <Þ@Þ ith distribution function 1 ß E(X) =! B1 ( BÑÞ Alternatively, X B S=X(= hen X is a continuous <Þ@Þ ith :Þ.Þ0Þ 0 X, E(X)= ' ' B0XÐBÑ. - " œ B0X( BÑ.BÞ When X L 2 # # ÐH Ñ, e have E(X Ñ œ llx # and e define the variance # of X to be Var(X)=E((X. X ) ), checking easily that Var(X) # # =E(X Ñ ÐE(X)) œ "average of the square square of the average". When Y is also in L # ÐY), the covariance of X and Y is defined to be cov(x,y)=e((x-. X)(Y. Y)) ith easy checking shoing that cov(x,y)=e(xy) E(X)E(Y)="average of the product product of the averages". When X and Y are independent, cov(x,y) = 0 (mean of the product =product of the means) but the converse isn't true. ( 333Ñ The basic inequalities. Let ( H, T, P) be a probability space. As is true for all normed vector spaces, hen X and Y are in L : ÐHÑß e have the triangle inequalityllx+y : Ÿ llx : lly : as ell as the scaling equation llcx : = c ll X :. For : œ #ß X,Y> =E(XY) defines the inner product of X and Y and the function <, > has the same algebraic properties as the dot product on 8 Þ Specifically, this function is symmetric # # and -bilinear from L ÐHÑ L ÐH Ñ into and satisfies the Cauchy- Schartz inequality: X

10 <X,Y> Ÿ X # ] # ith equality Í one X or Y is 0 or each is a non-zero scalar multiple of the other. (4) Actually the proof of (4) is essentially identical ith the proof for dot products. ; : When 1 Ÿ : ;ß L ÐHÑ L ÐHÑ ith X : Ÿ X ; for each X in L ; ÐHÑÞ : In Math 495, e on't bother ith L spaces for : Á " or 2; hen e get to stochastic calculus, e'll deal exclusively ith members of L # ÐHÑÞ 3. Stochastic Processes 3.1 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 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 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.Þ0 Þ = 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. 3.2 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 are so complicated that future predictions (conditional probabilities) based on current conditions are 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

11 topic in modern probability theory. Not surprisingly, the Markov processes here one can prove nice theorems and explicitly do calculations tend to be oversimplified limit versions of discrete real orld processes. 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 hen e use an object to construct other objects. In this vein, subscripts and superscripts ere used above to pass from an initial Ð8Ñ Ð_Ñ probabilty measure P on T to various other measures 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 feasible, 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 countably additive function on a certain 5 algebra 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 entering into the same 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! 4.2. Despite the above lament, one can use common sense to resolve many difficulties ith probability notations. Thus, although 0 X is the standard notation for the :Þ.Þ0 Þ of a continuous random variable, some authors rite 0 X rather than 1X for the distribution function of a discrete random variable. In a context here only discrete <Þ@Þ = or only continuous

12 = are under discussion, there's no problem. In the rare cases, here both types of <Þ@Þ = are being simultaneously discussed, it's best to reserve 0\ as a :Þ.Þ0 Þ symbol and revert to 1 \ for distribution functions of discrete <Þ@Þ =Þ Similarly, hen there are just too many different types of probability measures under discussion and calling each one of them P leads to a massive headache,, don't be afraid to go "against the tide" by inventing your on decorating scheme. Note that -Þ.Þ0 Þ = 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 and parametric statistics areas. Exactly the same comments as above apply--use common sense to avoid confusion.