LECTURE NOTES Course 6.041-6.431 M.I.T. FALL 2000 Introduction to Probability Dimitri P. Bertsekas and John N. Tsitsiklis Professors of Electrical Engineering and Computer Science Massachusetts Institute of Technology Cambridge, Massachusetts These notes are copyright-protected but may be freely distributed for instructional nonprofit pruposes.
Contents 1. Sample Space and Probability . . . . . . . . . . . . . . . . 1.1. Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Probabilistic Models . . . . . . . . . . . . . . . . . . . . . . . 1.3. Conditional Probability . . . . . . . . . . . . . . . . . . . . . 1.4. Independence . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5. Total Probability Theorem and Bayes’ Rule . . . . . . . . . . . . 1.6. Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7. Summary and Discussion . . . . . . . . . . . . . . . . . . . . 2. Discrete Random Variables . . . . . . . . . . . . . . . . . 2.1. Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Probability Mass Functions . . . . . . . . . . . . . . . . . . . 2.3. Functions of Random Variables . . . . . . . . . . . . . . . . . . 2.4. Expectation, Mean, and Variance . . . . . . . . . . . . . . . . . 2.5. Joint PMFs of Multiple Random Variables . . . . . . . . . . . . . 2.6. Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7. Independence . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8. Summary and Discussion . . . . . . . . . . . . . . . . . . . . 3. General Random Variables . . . . . . . . . . . . . . . . . 3.1. Continuous Random Variables and PDFs . . . . . . . . . . . . . 3.2. Cumulative Distribution Functions . . . . . . . . . . . . . . . . 3.3. Normal Random Variables . . . . . . . . . . . . . . . . . . . . 3.4. Conditioning on an Event . . . . . . . . . . . . . . . . . . . . 3.5. Multiple Continuous Random Variables . . . . . . . . . . . . . . 3.6. Derived Distributions . . . . . . . . . . . . . . . . . . . . . . 3.7. Summary and Discussion . . . . . . . . . . . . . . . . . . . . 4. Further Topics on Random Variables and Expectations . . . . . . 4.1. Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Sums of Independent Random Variables - Convolutions . . . . . . . iii
iv Contents 4.3. Conditional Expectation as a Random Variable . . . . . . . . . . . 4.4. Sum of a Random Number of Independent Random Variables . . . . 4.5. Covariance and Correlation . . . . . . . . . . . . . . . . . . . 4.6. Least Squares Estimation . . . . . . . . . . . . . . . . . . . . 4.7. The Bivariate Normal Distribution . . . . . . . . . . . . . . . . 5. The Bernoulli and Poisson Processes . . . . . . . . . . . . . . 5.1. The Bernoulli Process . . . . . . . . . . . . . . . . . . . . . . 5.2. The Poisson Process . . . . . . . . . . . . . . . . . . . . . . . 6. Markov Chains . . . . . . . . . . . . . . . . . . . . . . . 6.1. Discrete-Time Markov Chains . . . . . . . . . . . . . . . . . . 6.2. Classification of States . . . . . . . . . . . . . . . . . . . . . . 6.3. Steady-State Behavior . . . . . . . . . . . . . . . . . . . . . . 6.4. Absorption Probabilities and Expected Time to Absorption . . . . . 6.5. More General Markov Chains . . . . . . . . . . . . . . . . . . . 7. Limit Theorems . . . . . . . . . . . . . . . . . . . . . . . 7.1. Some Useful Inequalities . . . . . . . . . . . . . . . . . . . . . 7.2. The Weak Law of Large Numbers . . . . . . . . . . . . . . . . . 7.3. Convergence in Probability . . . . . . . . . . . . . . . . . . . . 7.4. The Central Limit Theorem . . . . . . . . . . . . . . . . . . . 7.5. The Strong Law of Large Numbers . . . . . . . . . . . . . . . .
Preface These class notes are the currently used textbook for “Probabilistic Systems Analysis,” an introductory probability course at the Massachusetts Institute of Technology. The text of the notes is quite polished and complete, but the prob- lems are less so. The course is attended by a large number of undergraduate and graduate students with diverse backgrounds. Acccordingly, we have tried to strike a bal- ance between simplicity in exposition and sophistication in analytical reasoning. Some of the more mathematically rigorous analysis has been just sketched or intuitively explained in the text, so that complex proofs do not stand in the way of an otherwise simple exposition. At the same time, some of this analysis and the necessary mathematical results are developed (at the level of advanced calcu- lus) in theoretical problems, which are included at the end of the corresponding chapter. The theoretical problems (marked by *) constitute an important com- ponent of the text, and ensure that the mathematically oriented reader will find here a smooth development without major gaps. We give solutions to all the problems, aiming to enhance the utility of the notes for self-study. We have additional problems, suitable for homework assignment (with solutions), which we make available to instructors. Our intent is to gradually improve and eventually publish the notes as a textbook, and your comments will be appreciated Dimitri P. Bertsekas bertsekas@lids.mit.edu John N. Tsitsiklis jnt@mit.edu v
1 Sample Space and Probability Contents 1.1. Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 3 1.2. Probabilistic Models . . . . . . . . . . . . . . . . . . . . p. 6 1.3. Conditional Probability . . . . . . . . . . . . . . . . . p. 16 1.4. Total Probability Theorem and Bayes’ Rule . . . . . . . . p. 25 1.5. Independence . . . . . . . . . . . . . . . . . . . . . . p. 31 1.6. Counting . . . . . . . . . . . . . . . . . . . . . . . p. 41 1.7. Summary and Discussion . . . . . . . . . . . . . . . . p. 48 1
2 Sample Space and Probability Chap. 1 “Probability” is a very useful concept, but can be interpreted in a number of ways. As an illustration, consider the following. A patient is admitted to the hospital and a potentially life-saving drug is administered. The following dialog takes place between the nurse and a concerned relative. RELATIVE: Nurse, what is the probability that the drug will work? NURSE: I hope it works, we’ll know tomorrow. RELATIVE: Yes, but what is the probability that it will? NURSE: Each case is different, we have to wait. RELATIVE: But let’s see, out of a hundred patients that are treated under similar conditions, how many times would you expect it to work? NURSE (somewhat annoyed): I told you, every person is different, for some it works, for some it doesn’t. RELATIVE (insisting): Then tell me, if you had to bet whether it will work or not, which side of the bet would you take? NURSE (cheering up for a moment): I’d bet it will work. RELATIVE (somewhat relieved): OK, now, would you be willing to lose two dollars if it doesn’t work, and gain one dollar if it does? NURSE (exasperated): What a sick thought! You are wasting my time! In this conversation, the relative attempts to use the concept of probability to discuss an uncertain situation. The nurse’s initial response indicates that the meaning of “probability” is not uniformly shared or understood, and the relative tries to make it more concrete. The first approach is to define probability in terms of frequency of occurrence, as a percentage of successes in a moderately large number of similar situations. Such an interpretation is often natural. For example, when we say that a perfectly manufactured coin lands on heads “with probability 50%,” we typically mean “roughly half of the time.” But the nurse may not be entirely wrong in refusing to discuss in such terms. What if this was an experimental drug that was administered for the very first time in this hospital or in the nurse’s experience? While there are many situations involving uncertainty in which the fre- quency interpretation is appropriate, there are other situations in which it is not. Consider, for example, a scholar who asserts that the Iliad and the Odyssey were composed by the same person, with probability 90%. Such an assertion conveys some information, but not in terms of frequencies, since the subject is a one-time event. Rather, it is an expression of the scholar’s subjective be- lief. One might think that subjective beliefs are not interesting, at least from a mathematical or scientific point of view. On the other hand, people often have to make choices in the presence of uncertainty, and a systematic way of making use of their beliefs is a prerequisite for successful, or at least consistent, decision
Sec. 1.1 Sets 3 making. In fact, the choices and actions of a rational person, can reveal a lot about the inner-held subjective probabilities, even if the person does not make conscious use of probabilistic reasoning. Indeed, the last part of the earlier dialog was an attempt to infer the nurse’s beliefs in an indirect manner. Since the nurse was willing to accept a one-for-one bet that the drug would work, we may infer that the probability of success was judged to be at least 50%. And had the nurse accepted the last proposed bet (two-for-one), that would have indicated a success probability of at least 2/3. Rather than dwelling further into philosophical issues about the appropri- ateness of probabilistic reasoning, we will simply take it as a given that the theory of probability is useful in a broad variety of contexts, including some where the assumed probabilities only reflect subjective beliefs. There is a large body of successful applications in science, engineering, medicine, management, etc., and on the basis of this empirical evidence, probability theory is an extremely useful tool. Our main objective in this book is to develop the art of describing un- certainty in terms of probabilistic models, as well as the skill of probabilistic reasoning. The first step, which is the subject of this chapter, is to describe the generic structure of such models, and their basic properties. The models we consider assign probabilities to collections (sets) of possible outcomes. For this reason, we must begin with a short review of set theory. 1.1 SETS Probability makes extensive use of set operations, so let us introduce at the outset the relevant notation and terminology. A set is a collection of objects, which are the elements of the set. If S is a set and x is an element of S, we write x ∈ S. If x is not an element of S, we write x /∈ S. A set can have no elements, in which case it is called the empty set, denoted by Ø. Sets can be specified in a variety of ways. If S contains a finite number of elements, say x1, x2, . . . , xn, we write it as a list of the elements, in braces: S = {x1, x2, . . . , xn}. For example, the set of possible outcomes of a die roll is {1,2,3,4,5,6}, and the set of possible outcomes of a coin toss is {H, T}, where H stands for “heads” and T stands for “tails.” If S contains infinitely many elements x1, x2, . . ., which can be enumerated in a list (so that there are as many elements as there are positive integers) we write S = {x1, x2, . . .},
4 Sample Space and Probability Chap. 1 and we say that S is countably infinite. For example, the set of even integers can be written as {0,2,−2,4,−4, . . .}, and is countably infinite. Alternatively, we can consider the set of all x that have a certain property P, and denote it by {x|x satisfies P}. (The symbol “|” is to be read as “such that.”) For example the set of even integers can be written as {k |k/2 is integer}. Similarly, the set of all scalars x in the interval [0,1] can be written as {x|0 ≤ x ≤ 1}. Note that the elements x of the latter set take a continuous range of values, and cannot be written down in a list (a proof is sketched in the theoretical problems); such a set is said to be uncountable. If every element of a set S is also an element of a set T, we say that S is a subset of T, and we write S ⊂ T or T ⊃ S. If S ⊂ T and T ⊂ S, the two sets are equal, and we write S = T. It is also expedient to introduce a universal set, denoted by Ω, which contains all objects that could conceivably be of interest in a particular context. Having specified the context in terms of a universal set Ω, we only consider sets S that are subsets of Ω. Set Operations The complement of a set S, with respect to the universe Ω, is the set {x ∈ Ω|x /∈ S} of all elements of Ω that do not belong to S, and is denoted by Sc. Note that Ωc = Ø. The union of two sets S and T is the set of all elements that belong to S or T (or both), and is denoted by S ∪ T. The intersection of two sets S and T is the set of all elements that belong to both S and T, and is denoted by S ∩ T. Thus, S ∪ T = {x|x ∈ S or x ∈ T}, S ∩ T = {x|x ∈ S and x ∈ T}. In some cases, we will have to consider the union or the intersection of several, even infinitely many sets, defined in the obvious way. For example, if for every positive integer n, we are given a set Sn, then ⋃∞ Sn = S1 ∪ S2 ∪ · · · = {x|x ∈ Sn for some n}, n=1 and ⋂∞ Sn = S1 ∩ S2 ∩ · · · = {x|x ∈ Sn for all n}. n=1 Two sets are said to be disjoint if their intersection is empty. More generally, several sets are said to be disjoint if no two of them have a common element. A collection of sets is said to be a partition of a set S if the sets in the collection are disjoint and their union is S.
Sec. 1.1 Sets 5 If x and y are two objects, we use (x, y) to denote the ordered pair of x and y. The set of scalars (real numbers) is denoted by ; the set of pairs (or triplets) of scalars, i.e., the two-dimensional plane (or three-dimensional space, respectively) is denoted by 2 (or 3, respectively). Sets and the associated operations are easy to visualize in terms of Venn diagrams, as illustrated in Fig. 1.1. Ω Ω Ω S S S T T T (a) (b) (c) Ω Ω Ω T S S T U S T U (d) (e) (f) Figure 1.1: Examples of Venn diagrams. (a) The shaded region is S ∩ T. (b) The shaded region is S ∪ T. (c) The shaded region is S ∩ Tc. (d) Here, T ⊂ S. The shaded region is the complement of S. (e) The sets S, T, and U are disjoint. (f) The sets S, T, and U form a partition of the set Ω. The Algebra of Sets Set operations have several properties, which are elementary consequences of the definitions. Some examples are: S ∪ T = T ∪ S, S ∪ (T ∪ U) = (S ∪ T) ∪ U, S ∩ (T ∪ U) = (S ∩ T) ∪ (S ∩ U), S ∪ (T ∩ U) = (S ∪ T) ∩ (S ∪ U), (Sc)c = S, S ∩ Sc = Ø, S ∪ Ω = Ω, S ∩ Ω = S. Two particularly useful properties are given by de Morgan’s laws which state that ( ) ( ) ⋃ c ⋂ ⋂ c ⋃ S = Sc , S = Sc . n n n n n n n n To establish the first law, suppose that x ∈ (∪nSn)c. Then, x /∈ ∪nSn, which implies that for every n, we have x /∈ Sn. Thus, x belongs to the complement
6 Sample Space and Probability Chap. 1 of every S , and x ∈ ∩ Sc . This shows that (∪ S )c ⊂ ∩ Sc . The converse n n n n n n n n inclusion is established by reversing the above argument, and the first law follows. The argument for the second law is similar. 1.2 PROBABILISTIC MODELS A probabilistic model is a mathematical description of an uncertain situation. It must be in accordance with a fundamental framework that we discuss in this section. Its two main ingredients are listed below and are visualized in Fig. 1.2. Elements of a Probabilistic Model • The sample space Ω, which is the set of all possible outcomes of an experiment. • The probability law, which assigns to a set A of possible outcomes (also called an event) a nonnegative number P(A) (called the proba- bility of A) that encodes our knowledge or belief about the collective “likelihood” of the elements of A. The probability law must satisfy certain properties to be introduced shortly. Probability L a w Event B P(B) Experiment P(A) Event A Sample Space Ω (Set of Outcomes) A B Events Figure 1.2: The main ingredients of a probabilistic model. Sample Spaces and Events Every probabilistic model involves an underlying process, called the experi- ment, that will produce exactly one out of several possible outcomes. The set of all possible outcomes is called the sample space of the experiment, and is denoted by Ω. A subset of the sample space, that is, a collection of possible
Sec. 1.2 Probabilistic Models 7 outcomes, is called an event.† There is no restriction on what constitutes an experiment. For example, it could be a single toss of a coin, or three tosses, or an infinite sequence of tosses. However, it is important to note that in our formulation of a probabilistic model, there is only one experiment. So, three tosses of a coin constitute a single experiment, rather than three experiments. The sample space of an experiment may consist of a finite or an infinite number of possible outcomes. Finite sample spaces are conceptually and math- ematically simpler. Still, sample spaces with an infinite number of elements are quite common. For an example, consider throwing a dart on a square target and viewing the point of impact as the outcome. Choosing an Appropriate Sample Space Regardless of their number, different elements of the sample space should be distinct and mutually exclusive so that when the experiment is carried out, there is a unique outcome. For example, the sample space associated with the roll of a die cannot contain “1 or 3” as a possible outcome and also “1 or 4” as another possible outcome. When the roll is a 1, the outcome of the experiment would not be unique. A given physical situation may be modeled in several different ways, de- pending on the kind of questions that we are interested in. Generally, the sample space chosen for a probabilistic model must be collectively exhaustive, in the sense that no matter what happens in the experiment, we always obtain an out- come that has been included in the sample space. In addition, the sample space should have enough detail to distinguish between all outcomes of interest to the modeler, while avoiding irrelevant details. Example 1.1. Consider two alternative games, both involving ten successive coin tosses: Game 1: We receive $1 each time a head comes up. Game 2: We receive $1 for every coin toss, up to and including the first time a head comes up. Then, we receive $2 for every coin toss, up to the second time a head comes up. More generally, the dollar amount per toss is doubled each time a head comes up. † Any collection of possible outcomes, including the entire sample space Ω and its complement, the empty set Ø, may qualify as an event. Strictly speaking, however, some sets have to be excluded. In particular, when dealing with probabilistic models involving an uncountably infinite sample space, there are certain unusual subsets for which one cannot associate meaningful probabilities. This is an intricate technical issue, involving the mathematics of measure theory. Fortunately, such pathological subsets do not arise in the problems considered in this text or in practice, and the issue can be safely ignored.
8 Sample Space and Probability Chap. 1 In game 1, it is only the total number of heads in the ten-toss sequence that mat- ters, while in game 2, the order of heads and tails is also important. Thus, in a probabilistic model for game 1, we can work with a sample space consisting of eleven possible outcomes, namely, 0,1, . . . ,10. In game 2, a finer grain description of the experiment is called for, and it is more appropriate to let the sample space consist of every possible ten-long sequence of heads and tails. Sequential Models Many experiments have an inherently sequential character, such as for example tossing a coin three times, or observing the value of a stock on five successive days, or receiving eight successive digits at a communication receiver. It is then often useful to describe the experiment and the associated sample space by means of a tree-based sequential description, as in Fig. 1.3. Sample Space Sequential Tree Pair of Rolls Description 1, 1 4 1 1, 2 1, 3 1, 4 3 2 2nd Roll Root Leaves 2 3 1 1 2 3 4 4 1st Roll Figure 1.3: Two equivalent descriptions of the sample space of an experiment involving two rolls of a 4-sided die. The possible outcomes are all the ordered pairs of the form (i, j), where i is the result of the first roll, and j is the result of the second. These outcomes can be arranged in a 2-dimensional grid as in the figure on the left, or they can be described by the tree on the right, which reflects the sequential character of the experiment. Here, each possible outcome corresponds to a leaf of the tree and is associated with the unique path from the root to that leaf. The shaded area on the left is the event {(1,4), (2,4), (3,4), (4,4)} that the result of the second roll is 4. That same event can be described as a set of leaves, as shown on the right. Note also that every node of the tree can be identified with an event, namely, the set of all leaves downstream from that node. For example, the node labeled by a 1 can be identified with the event {(1,1),(1,2),(1,3),(1,4)} that the result of the first roll is 1. Probability Laws Suppose we have settled on the sample space Ω associated with an experiment.
Sec. 1.2 Probabilistic Models 9 Then, to complete the probabilistic model, we must introduce a probability law. Intuitively, this specifies the “likelihood” of any outcome, or of any set of possible outcomes (an event, as we have called it earlier). More precisely, the probability law assigns to every event A, a number P(A), called the probability of A, satisfying the following axioms. Probability Axioms 1. (Nonnegativity) P(A) ≥ 0, for every event A. 2. (Additivity) If A and B are two disjoint events, then the probability of their union satisfies P(A ∪ B) = P(A) + P(B). Furthermore, if the sample space has an infinite number of elements and A1, A2, . . . is a sequence of disjoint events, then the probability of their union satisfies P(A1 ∪ A2 ∪ · · ·) = P(A1) + P(A2) + · · · 3. (Normalization) The probability of the entire sample space Ω is equal to 1, that is, P(Ω) = 1. In order to visualize a probability law, consider a unit of mass which is to be “spread” over the sample space. Then, P(A) is simply the total mass that was assigned collectively to the elements of A. In terms of this analogy, the additivity axiom becomes quite intuitive: the total mass in a sequence of disjoint events is the sum of their individual masses. A more concrete interpretation of probabilities is in terms of relative fre- quencies: a statement such as P(A) = 2/3 often represents a belief that event A will materialize in about two thirds out of a large number of repetitions of the experiment. Such an interpretation, though not always appropriate, can some- times facilitate our intuitive understanding. It will be revisited in Chapter 7, in our study of limit theorems. There are many natural properties of a probability law which have not been included in the above axioms for the simple reason that they can be derived from them. For example, note that the normalization and additivity axioms imply that 1 = P(Ω) = P(Ω ∪ Ø) = P(Ω) + P(Ø) = 1 + P(Ø), and this shows that the probability of the empty event is 0: P(Ø) = 0.
10 Sample Space and Probability Chap. 1 As another example, consider three disjoint events A1, A2, and A3. We can use the additivity axiom for two disjoint events repeatedly, to obtain ( ) P(A1 ∪ A2 ∪ A3) = P A1 ∪ (A2 ∪ A2) = P(A1) + P(A2 ∪ A3) = P(A1) + P(A2) + P(A3). Proceeding similarly, we obtain that the probability of the union of finitely many disjoint events is always equal to the sum of the probabilities of these events. More such properties will be considered shortly. Discrete Models Here is an illustration of how to construct a probability law starting from some common sense assumptions about a model. Example 1.2. Coin tosses. Consider an experiment involving a single coin toss. There are two possible outcomes, heads (H) and tails (T). The sample space is Ω = {H, T}, and the events are {H, T}, {H}, {T}, Ø. If the coin is fair, i.e., if we believe that heads and tails are “equally likely,” we should assign equal probabilities to the two possible outcomes and specify that( ) ( ) P {H} = P {T} = 0.5. The additivity axiom implies that ( ) ( ) ( ) P {H, T} = P {H} + P {T} = 1, which is consistent with the normalization axiom. Thus, the probability law is given by ( ) ( ) ( ) P {H, T} = 1, P {H} = 0.5, P {T} = 0.5, P(Ø) = 0, and satisfies all three axioms. Consider another experiment involving three coin tosses. The outcome will now be a 3-long string of heads or tails. The sample space is Ω = {HHH, HHT, HT H, HT T, T HH, T HT, T T H, T T T}. We assume that each possible outcome has the same probability of 1/8. Let us construct a probability law that satisfies the three axioms. Consider, as an example, the event A = {exactly 2 heads occur} = {HHT, HT H, T HH}.
Sec. 1.2 Probabilistic Models 11 Using additivity, the probability of A is the sum of the probabilities of its elements: ( ) ( ) ( ) ( ) P {HHT, HT H, T HH} = P {HHT} + P {HT H} + P {T HH} 1 1 1 = + + 8 8 8 3 = . 8 Similarly, the probability of any event is equal to 1/8 times the number of possible outcomes contained in the event. This defines a probability law that satisfies the three axioms. By using the additivity axiom and by generalizing the reasoning in the preceding example, we reach the following conclusion. Discrete Probability Law If the sample space consists of a finite number of possible outcomes, then the probability law is specified by the probabilities of the events that consist of a single element. In particular, the probability of any event {s1, s2, . . . , sn} is the sum of the probabilities of its elements: ( ) ( ) ( ) ( ) P {s1, s2, . . . , sn} = P {s1} + P {s2} + · · · + P {sn} . ( ) In the special case where the probabilities P {s1}), . . . ,P({sn} are all the same (by necessity equal to 1/n, in view of the normalization axiom), we obtain the following. Discrete Uniform Probability Law If the sample space consists of n possible outcomes which are equally likely (i.e., all single-element events have the same probability), then the proba- bility of any event A is given by Number of elements of A P(A) = . n Let us provide a few more examples of sample spaces and probability laws. Example 1.3. Dice. Consider the experiment of rolling a pair of 4-sided dice (cf. Fig. 1.4). We assume the dice are fair, and we interpret this assumption to mean
12 Sample Space and Probability Chap. 1 that each of the sixteen possible outcomes [ordered pairs (i, j), with i, j = 1,2,3,4], has the same probability of 1/16. To calculate the probability of an event, we must count the number of elements of event and divide by 16 (the total number of possible outcomes). Here are some event probabilities calculated in this way: ( ) P {the sum of the rolls is even} = 8/16 = 1/2, ( ) P {the sum of the rolls is odd} = 8/16 = 1/2, ( ) P {the first roll is equal to the second} = 4/16 = 1/4, ( ) P {the first roll is larger than the second} = 6/16 = 3/8, ( ) P {at least one roll is equal to 4} = 7/16. Sample Space Pair of Rolls 4 3 Event 2nd Roll {at least one roll is a 4} Probability = 7/16 2 1 1 2 3 4 1st Roll Event {the first roll is equal to the second} Probability = 4/16 Figure 1.4: Various events in the experiment of rolling a pair of 4-sided dice, and their probabilities, calculated according to the discrete uniform law. Continuous Models Probabilistic models with continuous sample spaces differ from their discrete counterparts in that the probabilities of the single-element events may not be sufficient to characterize the probability law. This is illustrated in the following examples, which also illustrate how to generalize the uniform probability law to the case of a continuous sample space.
Sec. 1.2 Probabilistic Models 13 Example 1.4. A wheel of fortune is continuously calibrated from 0 to 1, so the possible outcomes of an experiment consisting of a single spin are the numbers in the interval Ω = [0,1]. Assuming a fair wheel, it is appropriate to consider all outcomes equally likely, but what is the probability of the event consisting of a single element? It cannot be positive, because then, using the additivity axiom, it would follow that events with a sufficiently large number of elements would have probability larger than 1. Therefore, the probability of any event that consists of a single element must be 0. In this example, it makes sense to assign probability b − a to any subinterval [a, b] of [0,1], and to calculate the probability of a more complicated set by eval- uating its “length.”† This assignment satisfies the three probability axioms and qualifies as a legitimate probability law. Example 1.5. Romeo and Juliet have a date at a given time, and each will arrive at the meeting place with a delay between 0 and 1 hour, with all pairs of delays being equally likely. The first to arrive will wait for 15 minutes and will leave if the other has not yet arrived. What is the probability that they will meet? Let us use as sample space the square Ω = [0,1] × [0,1], whose elements are the possible pairs of delays for the two of them. Our interpretation of “equally likely” pairs of delays is to let the probability of a subset of Ω be equal to its area. This probability law satisfies the three probability axioms. The event that Romeo and Juliet will meet is the shaded region in Fig. 1.5, and its probability is calculated to be 7/16. Properties of Probability Laws Probability laws have a number of properties, which can be deduced from the axioms. Some of them are summarized below. Some Properties of Probability Laws Consider a probability law, and let A, B, and C be events. (a) If A ⊂ B, then P(A) ≤ P(B). (b) P(A ∪ B) = P(A) + P(B) − P(A ∩ B). (c) P(A ∪ B) ≤ P(A) + P(B). (d) P(A ∪ B ∪ C) = P(A) + P(Ac ∩ B) + P(Ac ∩ Bc ∩ C). † ∫ The “length” of a subset S of [0,1] is the integral dt, which is defined, for S “nice” sets S, in the usual calculus sense. For unusual sets, this integral may not be well defined mathematically, but such issues belong to a more advanced treatment of the subject.
14 Sample Space and Probability Chap. 1 y 1 M 1/4 0 1/4 1 x Figure 1.5: The event M that Romeo and Juliet will arrive within 15 minutes of each other (cf. Example 1.5) is { ∣∣ } M = (x, y) |x − y| ≤ 1/4, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 , and is shaded in the figure. The area of M is 1 minus the area of the two unshaded triangles, or 1 − (3/4) · (3/4) = 7/16. Thus, the probability of meeting is 7/16. These properties, and other similar ones, can be visualized and verified graphically using Venn diagrams, as in Fig. 1.6. For a further example, note that we can apply property (c) repeatedly and obtain the inequality ∑n P(A1 ∪ A2 ∪ · · · ∪ An) ≤ P(Ai). i=1 In more detail, let us apply property (c) to the sets A1 and A2 ∪ · · · ∪ An, to obtain P(A1 ∪ A2 ∪ · · · ∪ An) ≤ P(A1) + P(A2 ∪ · · · ∪ An). We also apply property (c) to the sets A2 and A3 ∪ · · · ∪ An to obtain P(A2 ∪ · · · ∪ An) ≤ P(A2) + P(A3 ∪ · · · ∪ An), continue similarly, and finally add. Models and Reality Using the framework of probability theory to analyze a physical but uncertain situation, involves two distinct stages. (a) In the first stage, we construct a probabilistic model, by specifying a prob- ability law on a suitably defined sample space. There are no hard rules to
Sec. 1.2 Probabilistic Models 15 A U c U B A B A B A B (a) (b) A B c U c U C c U A B C A B (c) Figure 1.6: Visualization and verification of various properties of probability laws using Venn diagrams. If A ⊂ B, then B is the union of the two disjoint events A and Ac ∩ B; see diagram (a). Therefore, by the additivity axiom, we have P(B) = P(A) + P(Ac ∩ B) ≥ P(A), where the inequality follows from the nonnegativity axiom, and verifies prop- erty (a). From diagram (b), we can express the events A ∪ B and B as unions of disjoint events: A ∪ B = A ∪ (Ac ∩ B), B = (A ∩ B) ∪ (Ac ∩ B). The additivity axiom yields P(A ∪ B) = P(A) + P(Ac ∩ B), P(B) = P(A ∩ B) + P(Ac ∩ B). Subtracting the second equality from the first and rearranging terms, we obtain P(A∪B) = P(A) +P(B)−P(A∩B), verifying property (b). Using also the fact P(A ∩ B) ≥ 0 (the nonnegativity axiom), we obtain P(A ∪ B) ≤ P(A) + P(B), verifying property (c) From diagram (c), we see that the event A ∪ B ∪ C can be expressed as a union of three disjoint events: A ∪ B ∪ C = A ∪ (Ac ∩ B) ∪ (Ac ∩ Bc ∩ C), so property (d) follows as a consequence of the additivity axiom.
Hiển thị 20 trang đầu. Tải tài liệu để xem đầy đủ.
Chưa có bình luận nào. Hãy là người đầu tiên!
Bình luận (0)
Chưa có bình luận nào. Hãy là người đầu tiên!