1.2 AXIOMATIC APPROACH TO PROBABILITY AND PROPERTIES OF PROBABILITY MEASURE 1.2 AXIOMATIC APPROACH TO PROBABILITY AND

Transcription

1 NTEL- robability ad Distributios MODULE 1 ROBABILITY LECTURE 2 Topics 1.2 AXIOMATIC AROACH TO ROBABILITY AND ROERTIES OF ROBABILITY MEASURE Iclusio-Exclusio Forula I the followig sectio we will discuss the oder approach to probability theory where we will ot be cocered with how probabilities are assiged to suitably chose subsets of Ω. Rather we will defie the cocept of probability for certai types of subsets Ω usig a set of axios that are cosistet with properties (i)-(iii) of classical (or relative frequecy) ethod. We will also study various properties of probability easures. 1.2 AXIOMATIC AROACH TO ROBABILITY AND ROERTIES OF ROBABILITY MEASURE We begi this sectio with the followig defiitios. Defiitio 2.1 (i) A set whose eleets are theselves set is called a class of sets. A class of sets will be usually deoted by script letters A, B, C,. For exaple A = 1, 1, 3, 2, 5, 6 ; (ii) Let C be a class of sets. A fuctio μ: C R is called a set fuctio. I other words, a real-valued fuctio whose doai is a class of sets is called a set fuctio. As stated above, i ay situatios, it ay ot be possible to assig probabilities to all subsets of the saple space Ω such that properties (i)-(iii) of classical (or relative frequecy) ethod are satisfied. Therefore oe begis with assigig probabilities to ebers of a appropriately chose class C of subsets of Ω (e.g., if Ω = R, the C ay be class of all ope itervals i R; if Ω is a coutable set, the C ay be class of all sigletos ω, ω Ω). We call the ebers of C as basic sets. Startig fro the basic sets i C assiget of probabilities is exteded, i a ituitively justified aer, to as ay subsets of Ω as possible keepig i id that properties (i)-(iii) of classical (or Dept. of Matheatics ad Statistics Idia Istitute of Techology, Kapur 1

2 NTEL- robability ad Distributios relative frequecy) ethod are ot violated. Let us deote by F the class of sets for which the probability assigets ca be fially doe. We call the class F as evet space ad eleets of F are called evets. It will be reasoable to assue that F satisfies the followig properties: (i) Ω F, (ii) A F A C = Ω A F, ad (iii) A i F, i = 1,2, A i F. This leads to itroductio of the followig defiitio. Defiitio 2.2 A siga-field (σ -field) of subsets of Ω is a class F of subsets of Ω satisfyig the followig properties: (i) Ω F; (ii) A F A c = Ω A F (closed uder copleets); (iii) A i F, i = 1, 2, A i F (closed uder coutably ifiite uios). Reark 2.1 (i) (ii) We expect the evet space to be a σ-field; Suppose that F is a σ-field of subsets of Ω. The, (a) φ F sice φ = Ω c c (b) E 1, E 2, F F sice = E c i ; (c) E, F F E F = E F c F ad E ΔF E F F E F; (d) E 1, E 2,, E F, for soe N, F ad F (take E +1 = E +2 = = φ so that = or E +1 = E +2 = = Ω so that = ); (e) although the power set of Ω Ω is a σ -field of subsets of Ω, i geeral, a σ-field ay ot cotai all subsets of Ω. Exaple 2.1 (i) (ii) (iii) (iv) (v) F = φ, Ω is a siga field, called the trivial siga-field; Suppose that A Ω. The F = A, A c, φ, Ω is a σ-field of subsets of Ω. It is the sallest siga-field cotaiig the set A; Arbitrary itersectio of σ-fields is a σ-field (see roble 3 (i)); Let C be a class of subsets of Ω ad let F α α Λ be the collectio of all σfields that cotai C. The F = F α α Λ is a σ-field ad it is the sallest σ-field that cotais class C (called the σfield geerated by C ad is deoted by σ(c)) (see roble 3 (iii)); Let Ω = R ad let J be the class of all ope itervals i R. The B 1 = σ J is called the Borel σ-field o R. The Borel σ-field i R k (deoted by B k ) is the Dept. of Matheatics ad Statistics Idia Istitute of Techology, Kapur 2

3 NTEL- robability ad Distributios (vi) σ-field geerated by class of all ope rectagles i R k. A set B B k is called a Borel set i R k ; here R k = {(x 1,, x k ): < x i <, i = 1,, k} deotes the k-diesioal Euclidea space; B 1 cotais all sigletos ad hece all coutable subsets of R a = a 1, a + 1 =1. Let C be a appropriately chose class of basic subsets of Ω for which the probabilities ca be assiged to begi with (e.g., if Ω = R the C ay be class of all ope itervals i R; if Ω is a coutable set the C ay be class of all sigletos ω, ω Ω). It turs out (a topic for a advaced course i probability theory) that, for a appropriately chose class C of basic sets, the assiget of probabilities that is cosistet with properties (i)-(iii) of classical (or relative frequecy) ethod ca be exteded i a uique aer fro C to σ(c), the sallest σ-field cotaiig the class C. Therefore, geerally the doai F of a probability easure is take to be σ(c), the σ-field geerated by the class C of basic subsets of Ω. We have stated before that we will ot care about how assiget of probabilities to various ebers of evet space F (a σ-field of subsets of Ω) is doe. Rather we will be iterested i properties of probability easure defied o evet space F. Let Ω be a saple space associated with a rado experiet ad let F be the evet space (a σ-field of subsets of Ω). Recall that ebers of F are called evets. Now we provide a atheatical defiitio of probability based o a set of axios. Defiitio 2.3 (i) Let F be a σ -field of subsets of Ω. A probability fuctio (or a probability easure) is a set fuctio, defied o F, satisfyig the followig three axios: (a) E 0, E F; (Axio 1: No egativity); (b) If E 1, E 2, is a coutably ifiite collectio of utually exclusive evets i. e., F, i = 1, 2,, E j = φ, i j the = 1=1 ; (Axio 2: Coutably ifiite additive) (c) Ω = 1 (Axio 3: robability of the saple space is 1). (ii) The triplet Ω, F, is called a probability space. Dept. of Matheatics ad Statistics Idia Istitute of Techology, Kapur 3

4 NTEL- robability ad Distributios Reark 2.2 (i) Note that if E 1, E 2, is a coutably ifiite collectio of sets i a σ-field F the F ad, therefore, ( ) is well defied; (ii) I ay probability space Ω, F, we have Ω = 1 (or φ = 0; see Theore 2.1 (i) proved later) but if A = 1 (or A = 0), for soe A F, the it does ot ea that A = Ω ( or A = φ) (see roble 14 (ii)). (iii) I geeral ot all subsets of Ω are evets, i.e., ot all subsets of Ω are eleets of F. (iv) Whe Ω is coutable it is possible to assig probabilities to all subsets of Ω usig Axio 2 provided we ca assig probabilities to sigleto subsets x of Ω. To illustrate this let Ω = ω 1, ω 2, or Ω = ω 1,, ω, for soe N ad let ω i = p i, i = 1, 2,, so that 0 p i 1, i = 1,2, (see Theore 2.1 (iii) below) ad p i = ω i = ω i = Ω = 1. The, for ay A Ω, A = p i. i:ω i A Thus i this case we ay take F = Ω, the power set of Ω. It is worth etioig here that if Ω is coutable ad C = ω ω Ω (class of all sigleto subsets of Ω) is the class of basic sets for which the assiget of the probabilities ca be doe, to begi with, the σ(c) = Ω (see roble 5 (ii)). (v) Due to soe icosistecy probles, assiget of probabilities for all subsets of Ω is ot possible whe Ω is cotiuu (e.g., if Ω cotais a iterval). Theore 2.1 Let Ω, F, be a probability space. The (i) φ = 0; (ii) F, i = 1, 2,., ad E j = φ, i j = (fiite additivity); (iii) E F, 0 E 1 ad E c = 1 E ; (iv) E 1, E 2 F ad E 1 E 2 E 2 E 1 = E 2 E 1 ad E 1 E 2 (ootoicity of probability easures); (v) E 1, E 2 F E 1 E 2 = E 1 + E 2 E 1 E 2. roof. (i) Let E 1 = Ω ad = φ, i = 2, 3,. The E 1 = 1, (Axio 3), F, i = 1, 2,, E 1 = ad E j = φ, i j. Therefore, Dept. of Matheatics ad Statistics Idia Istitute of Techology, Kapur 4

5 NTEL- robability ad Distributios 1 = E 1 = = (usig Axio 2) = 1 + φ i=2 i=2 φ = 0 φ = 0. (ii) Let = φ, i = + 1, + 2,. The F, i = 1, 2,, E j = φ, i j ad = 0, i = + 1, + 2,. Therefore, = = usig Axio 2 =. (iii) Let E F. The Ω = E E c ad E E C = φ. Therefore 1 = Ω = E E c = E + E c (usig (ii)) E 1 ad E c = 1 E (sice (E c ) [0,1]) 0 E 1 ad E c = 1 E. (iv) Let E 1, E 2 F ad let E 1 E 2. The E 2 E 1 F, E 2 = E 1 E 2 E 1 ad E 1 E 2 E 1 = φ. Dept. of Matheatics ad Statistics Idia Istitute of Techology, Kapur 5

6 NTEL- robability ad Distributios Figure 2.1 Therefore, E 2 = E 1 E 2 E 1 = E 1 + E 2 E 1 (usig (ii)) E 2 E 1 = E 2 E 1. As E 2 E 1 0, it follows that E 1 E 2. (v) Let E 1, E 2 F. The E 2 E 1 F, E 1 E 2 E 1 = φ ad E 1 E 2 = E 1 E 2 E 1. Figure 2.2 Therefore, E 1 E 2 = E 1 E 2 E 1 = E 1 + E 2 E 1 (usig (ii)) (2.1) Also E 1 E 2 E 2 E 1 = φ ad E 2 = E 1 E 2 E 2 E 1. Therefore, Dept. of Matheatics ad Statistics Idia Istitute of Techology, Kapur 6

7 NTEL- robability ad Distributios Figure 2.3 E 2 = E 1 E 2 E 2 E 1 = E 1 E 2 + E 2 E 1 (usig (ii)) E 2 E 1 = E 2 E 1 E 2 (2.2) Usig (2.1) ad (2.2), we get E 1 E 2 = E 1 + E 2 E 1 E Iclusio-Exclusio Forula Theore 2.2 Let Ω, F, be a probability space ad let E 1, E 2,, E F N, 2. The = S k,, k=1 where S 1, = ad, for k 2, 3,,, S k, = 1 k k. 1 i 1 < <i k roof. We will use the priciple of atheatical iductio. Usig Theore 2.1 (v), we have E 1 E 2 = E 1 + E 2 E 1 E 2 Dept. of Matheatics ad Statistics Idia Istitute of Techology, Kapur 7

8 NTEL- robability ad Distributios = S 1,2 + S 2,2, where S 1,2 = E 1 + E 2 ad S 2,2 = E 1 E 2. Thus the result is true for = 2. Now suppose that the result is true for 2, 3,, for soe positive iteger 2. The +1 = E +1 = + E +1 E +1 (usig the result for = 2) = + E +1 E +1 = S i, + E +1 E +1 usig the result for = (2.3) Let F i = E +1, i = 1,.. The E +1 = F i = T k, (agai usig the result for = ), (2.4) k=1 where T 1, = F i = E +1 ad, for k 2, 3,,, T k, = 1 k 1 F i1 F i2 F ik 1 i 1 <i 2 < <i k = 1 k k E +1 1 i 1 <i 2 < <i k. Dept. of Matheatics ad Statistics Idia Istitute of Techology, Kapur 8

9 NTEL- robability ad Distributios Usig (2.4) i (2.3), we get +1 = S 1, + E +1 + S 2, T 1, + + S, T 1, T,. Note that S 1, + E +1 = S 1,+1, S k, T k 1, = S k,+1, k = 2,3,,, ad T, = S +1,+1. Therefore, = S 1,+1 + S k,+1 = S k,+1. k=2 k=1 Reark 2.3 (i) Let E 1, E 2, F. The E 1 E 2 E 3 = E 1 + E 2 + E 3 E 1 E 2 + E 1 E 3 + E 2 E 3 + E 1 E 2 E 3 S 1,3 S 2,3 S 3,3 = p 1,3 p 2,3 + p 3,3, where p 1,3 = S 1,3, p 2,3 = S 2,3 ad p 3,3 = S 3,3. I geeral, = p 1, p 2, + p 3, p,, where p i, = S i,, S i,, if i is odd, i = 1, 2,. if i is eve (ii) We have 1 E 1 E 2 = E 1 + E 2 E 1 E 2 E 1 E 2 E 1 + E 2 1. The above iequality is kow as Boferroi s iequality. Dept. of Matheatics ad Statistics Idia Istitute of Techology, Kapur 9