Probability Theory: Understanding Probability Spaces and Random Variables, Study notes of Stochastic Processes

Download Probability Theory: Understanding Probability Spaces and Random Variables and more Study notes Stochastic Processes in PDF only on Docsity! ECE 6010 Lecture 1 – Introduction; Review of Random Variables Readings from G&S: Chapter 1. Section 2.1, Section 2.3, Section 2.4, Section 3.1, Section 3.2, Section 3.5, Section 4.1, Section 4.2, Section 4.4, Section 4.5 Why study probability? 1. Communication systems. Noise. Information. 2. Control systems. Noise in observations. Noise in interference. 3. Computer systems. Random loads. Networks. Random packet arrival times. Probability can become a powerful engineering tool. One way of viewing it is as “quantified common sense.” Great success will come to those whose tool is sharp! Set theory Probability is intrinsically tied to set theory. We will review some set theory concepts. We will use c to denote complementation of a set with respect to its universe. – union: A B is the set of elements that are in A or B.
– intersection: A
B is the set of elements that are in A and B. We will also denote this as AB. a  A: a is an element of the set A. A  B: A is a subset of B. A  B: A  B and B  A. Note that A Ac   (where  is the universe). Notation for some special sets: R – set of all real numbers Z – set of all integers Z  – set of all positive integers N – set of all natural numbers, 0,1,2,…, Rn – set of all n tuples of real numbers C – set of complex numbers Definition 1 A field (or algebra) of sets is a collection of sets that is closed under comple- mentation and finite union. 2 That is, if F is a field and A  F , then Ac must also be in F (closed under complemen- tation). If A and B are in F (which we will write as A, B  F) then A B  F . Note: the properties of a field imply that F is also closed under finite intersection. (DeMorgan’s law: AB  (Ac Bc)c) Definition 2 A σ -field (or σ -algebra) of sets is a field that is also closed under countable unions (and intersections). 2 What do we mean by countable? • A set with a finite number in it is countable. • A set whose elements can be matched one-for-one with Z is countable (even if it has an infinite number of elements!) Are there non-countable sets? Note: For any collection F of sets, there is a σ -field containing F , denoted by σ(F). This is called the σ -field generated by F . ECE 6010: Lecture 1 – Introduction; Review of Random Variables 2 Definition of probability We now formally define what we mean by a probability space. A probability space has three components. The first is the sample space, which is the collection of all possible outcomes of some experiment. The outcome space is frequently denoted by . Example 1 Suppose the experiment involves throwing a die.  
1, 2, 3, 4, 5, 6  2 We deal with subsets of . For example, we might have an event which is “all even throws of the die” or “all outcomes  4”. We denote the collection of subsets of interest as F . The elements of F (that is, the subsets of ) are called events. F is called the event class. We will restrict F to be a σ -field. Example 2 Let 
1, 2, 3, 4, 5, 6  , and letF 
 1, 2, 3, 4  , 5, 6  , 2, 4, 6  , 1, 3, 5  , ...  (What do we need to finish this off?) 2 Example 3  
1, 2, 3  . We could take F as the set of all subsets of : F 
 , 1  , 2  , 3  , 1, 2  , 1, 3  , 2, 3  , 1, 2, 3  This is frequently denoted as F  2, and is called the power set of . 2 Example 4   R. F is restricted to something smaller than all subsets of . (So that probabilities can be applied consistently.) F could be the smallest σ -field which contains all intervals of R. This is called the Borel field, B. 2 The tuple (,F) is called a pre-probability space (because we haven’t assigned prob- abilities yet). This brings us to the third element of a probability space: Given a pre- probability space (,F), a probability distribution or a measure on (,F) is a mapping P from F to R (which we will write: P : F  R) with the properties: • P()  1 (this is a normalization that is always applied for probabilities, but there are other measures which don’t use this) • P(A) 0 A  F . (Measures are nonnegative) • If A1, A2, . . .  F such that Ai A j   for all i  j (that is, the sets are “disjoint” or “mutually exclusive”) then P(  i  1 Ai)   i  1 P(Ai ). This is called the σ -additive, or additive, property. These three properties are called the axioms of probability. The triple (,F, P) is called a probability space. •  tells what individual outcomes are possible • F tells what sets of outcomes — events — are possible. • P tells what the probabilities of these events are. Some properties of probabilities (which follow from the axioms): ECE 6010: Lecture 1 – Introduction; Review of Random Variables 5 2 Is independent the same as disjoint? Note: For P(B) > 0, if A and B are independent then P(A  B)  P(A). (Since they are independent, B can provide no information about A, so the probability remains unchanged. If P(B)  0, then B is independent of A for any other event A  F . (Why?). Definition 5 A1, . . . , An  F are independent if for each k  2, . . . , n  and each subset i1, . . . , ik  of 1, . . . , n  , P(
k j  1 Ai j )  k j  1 P(Ai j ). 2 Example 7 Take n  3. Independent if: P(A1 A2)  P(A1)P(A2) P(A1 A3)  P(A1)P(A3) P(A2 A3)  P(A2)P(A3) and P(A1 A2 A3)  P(A1)P(A2)P(A3). 2 The next idea is important in a lot of practical problem of engineering interest. Definition 6 A1 and A2 are conditionally independent given B  F if P(A1 A2  B)  P(A1  B)P(A2  B) 2 (draw picture to illustrate the idea). Random variables Up to this point, the outcomes in  could be anything: they could be elephants, computers, or mitochondria, since  is simple expressed in terms of sets. But we frequently deal with numbers, and want to describe events associated with sets of numbers. This leads to the idea of a random variable. Definition 7 Given a probability space (,F, P), a random variable is a function X mapping  to R. (That is, X :   R), such that for each a  R, ω   : X (ω)  a   F . 2 A function X :   R such that w   : X (ω)  a   F , that is, such that the events involved are in F , is said to be measurable with respect to F . That is, F is divided into sufficiently small pieces that the events in it can describe all of the sets associated with X . Example 8 Let   1, 2, 3, 4, 5, 6  , F 
 1, 2, 3  , 4, 5, 6  ,  ,   . Define X (ω) 
0 ω odd 1 ω otherwise Then for X to be a random variable, we must have ω    X (ω)  a  ECE 6010: Lecture 1 – Introduction; Review of Random Variables 6 to be an event in F . ω    X (ω)  a  
   a < 0 1, 3, 5  0  a < 1  a 1 So X is not a random variable — it not measurable with respect to F . The field F is too “coarse” to measure if ω is odd. On the other hand, let us now define X (ω) 
0 ω  3 1 ω > 3 Is this a random variable? 2 We observe that a random variable cannot generate partitions of the underlying sample space which are not events in the σ -field F . Another way of saying that X is measurable: For any B  B, ω   : X (ω)  B   F . Recalling the idea of Borel sets B associated with the real line, we see that a random variable is a measurable function from (,F) to (R,B): X : (,F)  (R,B). We will abbreviate the term random variable as r.v.. We will use a notational shorthand for random variables. Definition 8 Suppose (,F, P) is a probability space and X : (,F)  (R,B). For each B  B we define P(X  B)  P( ω   : X (ω)  B  ). 2 By this definition, we can identify a new probability space. Using (R,B) as the pre- probability space, we use the measure PX (B)  P(X  B). So we get the probability space (R,B, PX ). As a matter of practicality, if the sample space is R, with the Borel field, most mappings to R will be random variables. To summarize: (,F, P) X  (R,B, PX ) where PX (B)  P( ω    X (ω)  B  ) for B  B. Distribution functions The cumulative distribution function (cdf) of an r.v. X is defined for each a  R as FX (a)  P(X  a)  P( ω    X (ω)  a  )  PX ((  , a]) Properties of cdf: ECE 6010: Lecture 1 – Introduction; Review of Random Variables 7 1. FX is non-decreasing: If a < b then FX (a)  FX (b). 2. lima  FX (a)  1 3. lima 
FX (a)  0. 4. FX is right-continuous: limb  a  FX (b)  FX (a). Draw “typical” picture. These four properties completely characterize the family of cdfs on the real line. Any function which satisfies these has a corresponding probability distribution. 5. For b > a: P(a < X  b)  FX (b) FX (a). 6. P(X  a0)  FX (a0) lima  a 0 Fx (a). Thus, if FX is continuous at a0, P(X  a0)  0. From these properties, we can assign probabilities to all intervals from knowledge of the cdf. Thus we can extend this to all Borel sets. Thus FX determines a unique probability distribution on (R,B), so FX and PX are uniquely related. Pure types of r.v.s 1. Discrete r.v.s — an r.v. whose possible values can be enumerated. 2. Continuous r.v.s — an r.v. whose distribution function can be written as the (regular) integral of another function 3. Singular, but not discrete — Any other r.v. Discrete r.v.s A random variable that can take on at most a countable number of possible values is said to be a discrete r.v.: X :   x1, x2, . . .  Definition 9 For a discrete r.v. X , we define the probability mass function (pmf) (or discrete density function) by pX(a)  P(X  a) a  R where pX(a)  0 if a  xi for any r.v. outcome xi . 2 Properties of pmfs: 1. Nonnegativity: pX(a) 
0 a  x1, x2, . . .  0 else