Basic probability theory ppt. The spinner at the right is spun twice.

Go Green With Knowledge! Get 30% off on Annual Courses Probability tells us how often some event will happen after many repeated trials. What probability theory is for. Axiom 2: P(Ω) = 1. Dependent and independent events. The probability of an event is between 0 and 1. 1 Basic Definitions. P (¬A) = probability of a not happening event. regardless of the value the other r. May 12, 2017 · The probability of event A =. Probabilities can but need not be rounded. 1 Basic De nitions Trials? Probability is concerned with the outcome of tri-als. probability definition, probability theorem, addition theorem , multiplication theorem solved problems Aug 21, 2014 · Some basic Probability Rules Rule 1: The probability of any event E is a number between and including 0 and 1. 267 or rounded to . • An event is said to occur if one of the outcomes contained within the event occurs. THIAGARAJAN ASSOCIATE PROFESSOR OF MATHEMATICS ST JOSEPH'S COLLEGE TRICHIRAPPALLI Uncertainty in AI Outline: Introduction Basic Probability Theory Probabilistic Reasoning Why should we use probability theory? Probability. A classic example of a probabilistic This presentation guide her thrown Basic Probability Theory and Statistics, those are Per Experiment, Sample Distance, Irregular Variables, Probability, Conditional Probability, Variance, Probability Distribution, Joint Probability Distribution, Conditional Probability Distribution (CPD) both Factor. Jeff draws balls from the jar below. ppt), PDF File (. 1 Elements of probability In order to define a probability on a set we need a few basic elements, Sample space This presentation guide you through Basic Probability Theory and Statistics, those are Random Experiment, Sample Space, Random Variables, Probability, Conditional Probability, Variance, Probability Distribution, Joint Probability Distribution, Conditional Probability Distribution (CPD) and Factor. The are various other May 17, 2018 · Dr. What is the probability P 1 that this is a man? If you pick two persons randomly, what is the probability P 2 that these are a man and woman Answer: You have the possible outcomes: (M), (W1), (W2) so P 1 = A Tutorial on Probability Theory 1. Probability of an Event Not Occurring: If you want to find the probability of an event not happening, you subtract the probability of the event happening from 1. that students are already familiar with basic probability theory. subjective probability. • Probability and Statistics for Engineering and the Sciences by Jay L. For more topics stay tuned with Learnbay. The meaning of probability is basically the extent to which something is likely to happen. Basic Probability Basic Concepts Random Experiment is a process leading to at least Oct 4, 2021 · 6. The mathematical theory of probability is very sophisticated, and delves into a branch of analysis known as measure theory. The most popular theory posits that the dinosaurs were killed by the ensuing environmental catastrophe. Event: Each possible outcome of a variable is called an event. Theory of Probability, Lecture Slide 38. Therefore, in this third part, we assume that the reader is familiar with the Probability. Brownian Motion (PDF) 37. 6. 1. morley; Category. Upload. Sc , M. ? If the trial consists of ipping a coin twice, the Title: Basic Concepts of Discrete Probability 1 Basic Concepts of Discrete Probability (Theory of Sets Continuation) 2 Functions. 639 views • 47 slides Many basic probability problems are counting problems. 2. 141 likes • 54,512 views. It discusses common probability terms like experiment, outcome, sample space, event, and sample point. i. Classical probability • P (E) = # of outcomes in E___________ • Total # of outcomes in sample space • You roll a six-sided die. The occurrence of R is difficult to predict — we have all been victims of wrong forecasts Apr 3, 2019 · Probability Theory. Take for example the probability of rolling a dice and getting a 2 for the first time and for the second time. Here there are no possible outcomes in the event. element = an object in a set, denoted by a lower case Latin letter We say “ is an element of ,” “ is in ,” or “ belongs to ,” denoted as . Probability theory or probability calculus is the branch of mathematics concerned with probability. The probability that an event does not occur is 1 minus the probability that it does occur. The text-books listed below will be useful for other courses on probability and statistics. (IITK) Basics of Probability and Probability Distributions 7. 12. • Basic Properties of Probability – Assume that S is a sample space for some experiment and E is an event in S. potx file. De- Oct 4, 2021 · This presentation guide you through Basic Probability Theory and Statistics, those are Random Experiment, Sample Space, Random Variables, Probability, Conditional Probability, Variance, Probability Distribution, Joint Probability Distribution, Conditional Probability Distribution (CPD) and Factor. However, we would appreciate a citation where possible. This document provides an overview of probability concepts including: - Probability is a numerical measure of the likelihood of an event occurring, ranging from 0 (impossible) to 1 (certain). 00. 041SC Probabilistic …. Description: This file contains the information regarding theory of probability, lecture slide 1. Does the fossil record confirm that the disappearance of the dinosaurs was suitably instantaneous? Find the probability of throwing an 8 on a normal die. We will give a very quick review; some references for further reading appear at the end of the chapter. Title: Basic Probability 1 Basic Probability. This is the basic probability theory, which is also used in the probability distribution, where you will learn the possibility of outcomes for a random experiment. A scientifically based process composed of 4. Jun 23, 2017 · Probability Mmedsc Hahm. Frequency Theory. the empty set. motivation game theory gambling ; apparatuses coins dice ; playing therewith Theory of Probability, Lecture Slide 37. It is extremely useful in predicting and evaluating system performance. Types of probability • Three types of probability: • 1. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. ? Trials are also called experiments or observa-tions (multiple trials). The outcome of a random event cannot be determined before it occurs, but it may be any one of several possible outcomes. pptx or . The Probability of an Event because the number of outcomes in an event must be less than or equal to the number of outcomes in the sample space, the probability of an event will always be a number between 0 and 1, that is, 0≤P (E)≤1. It refers to the frequency at which some events or experiments occur. u k location Bioscience Building (New Biology), 2. Bayes' theorem is also known as Bayes' rule, Bayes' law, or Bayesian reasoning, which determines the probability of an event with uncertain knowledge. Oct 26, 2014 · Probability basics and bayes' theorem. 4) 1. Create a sample space with equally likely outcomes for a spinner with 4 sections numbered 1,2,3,4. It was presented by P. The probability that a large earthquake will occur on the San Andreas Fault in Oct 4, 2016 · It defines probability as the likelihood of an event occurring, expressed as a number between 0 and 1. yp(X;Y = y); p(Y) = P. For example: consider that you have two bags, named A and B, each containing 10 red balls and 10 black balls. Queuing theory has been used for operations research, manufacturing and systems analysis. Consider, as an example, the event R “Tomorrow, January 16th, it will rain in Amherst”. ppt, Subject Mathematics, from Indian Institute of Technology, Kharagpur, Length: 46 pages, Preview: DR. Therefore, it can be copied and reproduced without limitation. P (A and B) = P (A) x P (B) or P (A∩ 𝐵) = 𝑃 (𝐴) ∙ 𝑃 (𝐵) 21. Sample space (space of events). The probability that a selection of 6 numbers wins the National Lottery Lotto jackpot is 1 in 49 6 =13,983,816, or 7:15112 10 8. set = a collection of objects, denoted by an upper case Latin letter Example: . 1 Jan 3, 2020 · Basic Laws of Probability • The additive law of probabilities: given a set of mutually exclusive events, the probability of occurrence of one event or another event is equal to the sum of their separate probabilities. A probability of 1 is equivalent to 100% certainty. Sample Spaces . Jun 11, 2012 · The document provides an overview of key probability concepts including: 1. random variable probability distribution 5. The word probability has several meanings in ordinary conversation. This document discusses basic concepts of probability taught in a class. B ∪ C = "Sum of two dice is divisible by 3 or 4". Some properties of the operation of union: (i)A∪B = B∪A (Commutative law) (ii)A∪(B∪C) = (A∪B)∪C (Associative law) A. 37; download. OCW is open and available to the world and is a permanent MIT activity. Presentation on theme: "Basic Probability Concepts"— Presentation transcript: 1 Basic Probability Concepts Objective. Theory of Probability, Lecture Slide 1. Step 3: Since the event we’re interested in is the one consisting of rolls of 4, 5, or 7. pdf), Text File (. The higher the probability of an event, the more likely it is that the event will occur. ’s: p(X) = P. C = "Sum of two dice is divisible by 4". Events and their probability 4. 1 Basic probability theory Professor Jørn Vatn . ). N. The document defines key probability terms like random experiments, sample spaces, sample points, events, and the different types of events. 2 Probability Probability is a numerical measurement of likelihood of an event. Traditional queuing theory problems refer to customers visiting a store, analogous to requests arriving at a device. paired t - Test 7. Jul 16, 2017 · In words, we divide probability of both Rain and Sunny by the probability of a Sunny weather. Basic probability theory • Definition: Real-valued random variableX is a real-valued and measurable function defined on the sample space Ω, X: Ω→ ℜ – Each sample point ω ∈ Ω is associated with a real number X(ω) • Measurabilitymeans that all sets of type belong to the set of events , that is {X ≤ x} ∈ The basic formula for computing binomial coe cients is n k = n! k!(n k)!: (1. Events are called independent if the probability of one event does not influence the other in any way. EE 178/278A: Basic Probability Page 1–5 Elements of Probability • Probability theory provides the mathematical rules for assigning probabilities to outcomes of random experiments, e. Jul 30, 2012 · Basic Probability. I have taught students like these in courses on NLP and computational cognitive science, Basic Probability 1. ac. These terms are used in classical probability theory, but are also applicable in contemporary probability theory based on the theory of sets. Equivalently, we can represent the subset via a A Tutorial on Probability Theory 1. 1: The Basics of Probability Theory. It discusses key probability concepts like: - Probability is defined as the number of desired outcomes divided by the total number of possible outcomes and must be between 0 and 1. This document provides an overview of probability theory and concepts. Theorem 2 ( ) 10 ≤≤ AP Theorem 3 (a): Addition theorem of probability or theorem of Total Probability. Last Lecture (PDF) This section provides the schedule of lecture topics and the lecture slides used for each session. It covers counting sample points using tree diagrams, multiplication rules, permutations, and combinations. If you randomly pick up the ball from any bag (without Apr 6, 2019 · The probability of an event is obtained by summing the probabilities of the outcomes contained within the event A. We will assign a real number P(A) to every event A, called the probability of A. 3. Figure 7. it is the sum of the PMF table along the Jul 16, 2014 · BASIC NOTIONS OF PROBABILITY THEORY. Go deeper with your understanding of probability as you learn about theoretical, experimental, and compound probability, and investigate permutations, combinations, and more! p(X = x;Y = y)dxdy = 1. are disjoint then. - Tree diagrams can show all Course Description. Axiom 3: If A1,A2, . The occurrence of R is difficult to predict — we have all been victims of wrong forecasts To find the union of two given sets A and B is a set which consists of all the elements of A and all the elements of B such that no element is repeated. If a probability is determined to be, say, P (A) = . Additionally, it explains concepts Oct 17, 2019 · probability. The document summarizes key concepts in probability and statistics as they relate to biostatistics and medical research. The easiest way to illustrate the concept is with an example. To qualify as a probability, P must satisfy three axioms: Axiom 1: P(A) ≥ 0 for every A. The probability of any event is a number between zero and one. 4 red. Coin tossing Dice rolling (craps) Card games (blackjack, poker, etc. He draws two balls, this time with replacement. 2 yellow. All the Probability PowerPoint templates are natively built in PowerPoint, using placeholders on the slide master, color palettes, and other features in PowerPoint, and can contain layouts, theme colors, theme fonts, theme effects, background styles, and even content (according to Microsoft Now, each of the 36 ordered pairs in the table represent an equally likely outcome. Match case Limit results 1 per page. A mathematical approach to making the notion of “chance” rigorous. Step 2: To make our analysis easier, let’s replace each ordered pair with the sum (Figure 7. 0 ≤ pr (A) ≤ 1 2. &ndash; A free PowerPoint PPT presentation Classical Probability (Equally Likely Outcomes): To find the probability of an event happening, you divide the number of ways the event can happen by the total number of possible outcomes. Probability theory helps explain genetic theory. Basic Probability Concepts: Sample Spaces and Events, Simple Probability, and Joint Probability, Conditional Probability Bayes ’ Theorem Probability Distribution. The three main approaches to defining probability: classical, relative frequency, and subjective. Jan 25, 2023 · Theorems on Probability: Learn the basic rules of probability, types and theorems, with solved examples from this page. In these notes, we provide a basic treatment of probability that does not address these finer details. It also covers different types of probability like classical, statistical, and subjective probability. B = "Sum of two dice is divisible by 3". 267, it can be reported as . Jan 14, 2011 · probability ppt. Probability of Independent Events Example 1. ppt - Free download as PDF File (. The sum of these probabilities is 1. The sum of the probabilities of all possible outcomes is 1 or 100%. Best applied to processes which can be repeated many times , independently , and under the same conditions . To Nov 1, 2014 · If the outcomes in a sample space are not equally likely, then you must add the probabilities of all the individual outcomes in E. Nov 20, 2023 · Introduction to Basics of Probability Theory. Oct 5, 2009 · 2. For more topics stay adjusted with Learnbay. lecture Probability_theory. Download File. Probability is always between 0 and 1, with 1 being a certain event and 0 being an impossible event. Topics. pdf. takes For discrete r. empirical probability • 3. Random experiments, sample spaces, events, and the classification of events as simple, mutually exclusive, independent, and exhaustive. Feb 15, 2024 · Document Probability theory. Bayes' theorem was named after the British mathematician Thomas Bayes. 3) Note the important identity n k = n n k : (1. Hence the probability of throwing an 8 is 0 6 =0. Theory of Probability, Lecture Slide 39. The only other thing that I need to point out is that probability theory allows you to talk about non elementary events as well as elementary ones. It also covers marginal, conditional, and joint probabilities as well as the multiplication rule, addition rule, and how they apply to dependent and independent events. We can find the probability of an uncertain event by using the below formula. These tools underlie important advances in many fields, from the basic sciences to engineering and management. Random experiments 2. 0 < P (E) < 1 2. Feb 26, 2014 · BASIC NOTIONS OF PROBABILITY THEORY. Sample space: The collection of all possible events is called sample space. v. Probability is the measure of the likelihood that an event will occur in a Random Experiment. Manjunath from Indira College of Education in Tumkur. M. Example: Assume there are 1 man and 2 women in a room. It discusses: 1. york. txt) or view presentation slides online. Queuing Theory Queuing theory is the mathematics of waiting lines. Sample space 3. 1 Events and Complements (2/6) • A sample space consists of eight outcomes with a probability value. It also defines basic terminology like experiments, trials, outcomes, and events. Yet many students with backgrounds in linguistics, psychology, or other social sciences (and even some computer science students) have very little exposure to probability theory. Union, Intersection: For the two dice example, if. 2 Sampling with replacement Let Ibe a set with nelements and let Mbe a set with melements. empty set = null set = a set with no elements, denoted by space = the set with all the Probability theory is the mathematical framework that allows us to analyze chance events in a logically sound manner. The tools of probability theory, and of the related field of statistical inference, are the keys for being able to analyze and make sense of data. Examples are provided to illustrate different A PowerPoint template is a pattern or blueprint for your slides that you save as a . The probability that a fair coin will land heads is 1=2. Collection of all Possible Outcomes e. Sample Space (S)? Set of all possible elementary outcomes of a trial. Title: Basic principles of probability theory 1 Name Garib Murshudov (when asking questions Garib is sufficient) e-mail garib_at_ysbl. Probability and Uncertainty Probability measures the amount of uncertainty of an event: a fact whose occurrence is uncertain. ppt from DEPARTMENT OF MATHEMATICS 271 at University of Dar es salaam. The textbook for this subject is Bertsekas, Dimitri, and John Tsitsiklis. 0 ≤ P (E) ≤ 1 Rule 2: If an event E cannot occur, which means the event E is not in the sample space, then its probability is 0. Definitions of probability, including the frequency and subjective concepts. Ec = "Sum of two dice different from 7". Basic Probability. An extremely large meteor crashed into the earth at the time of the disappearance of the dinosaurs. View Notes - 1. Probability. Comparison of results of above tests and is useful for B. 11. - An experiment generates outcomes that make up the sample space. Basic Probability Theory. What probability theory is for • Suppose you’ve already texted the characters “There in a minu” • You’d like your mobile phone to guess the most likely completion of “minu” rather than MINUET or MINUS or MINUSCULE • In other words, you’d like your mobile phone to know that given what you’ve texted so far, MINUTE is more likely than Jan 13, 2015 · This document provides an introduction to probability and its applications in daily life. 2. It helps finding all the possible values a random variable can take between the minimum and maximum statistically possible values. P (¬A) + P (A) = 1. We hope that the reader has seen a little basic probability theory previously. 672 kB. The actual outcome is considered to be determined by chance. e. Typically these axioms formalise Jan 1, 2013 · The most significant fundamental concepts of the theory of probability applied in structural reliability include: Experiment; Random event; and. Probability simply talks about how likely is the event to occur, and its value always lies between 0 and 1 (inclusive of 0 and 1). B ∩ C = BC = "Sum of two dice is divisible by 3 and 4". Southern Range, Berhampur, Odisha. Oct 21, 2020 · This presentation covered the following topics : 1. The symbol for denoting union of sets is ‘∪‘. Part I: The Fundamentals. Jan 8, 2024 · Each event has some probability of occurring: this probability is a number between 0 to 1. The sum of the probabilities of all possible outcomes in a sample space is 1. There are three types of probability: theoretical, experimental, and subjective. xp(X = x;Y) For discrete r. Oct 24, 2014 · Basic Probability. More Brownian Motion (PDF) 38. Marginal Probability Distribution. 101 likes • 21,457 views. Events with probability close to zero are less likely to occur. The probability of a specified event is the chance or likelihood that it will occur. t - Test 6. The spinner at the right is spun twice. Jun 13, 2024 · probability theory, a branch of mathematics concerned with the analysis of random phenomena. P (E) = 0 Rule 3: If an event E is certainly occur, then its probability is 1. List all outcomes: May 22, 2015 · Probability is the mathematics of chance that describes the likelihood of events. Oct 26, 2014 •. 1. Probability theory pro vides a mathematical foundation to concepts such as Òproba-bilityÓ, ÒinformationÓ, Òbelief Ó, ÒuncertaintyÓ, Òcon Þ denceÓ, ÒrandomnessÓ, Òv ari-abilityÓ, ÒchanceÓ and ÒriskÓ. severity of the consequences of exposure to a. Then. This number is always between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. Probability is a number between 0 and 1. You need at most one of the three textbooks listed below, but you will need the statistical tables. BASIC NOTIONS OF PROBABILITY THEORY. Dec 31, 2018 · This document provides an overview of teaching basic probability and probability distributions to tertiary level teachers. It can be written as a fraction, decimal, percent, or ratio between 0 and 1. Intuitively, the probability distribution of one r. Jul 31, 2012 · Probability concept and Probability distribution. You should be familiar with the basic tools of the gambling trade: a coin, a (six-sided) die, and a full deck of 52 cards. You pick a person randomly. Classical probability • 2. Let’s shade those in (Figure 7. In probability theory, it relates the conditional probability and marginal probabilities of two random events. P (S) = 1. This resource is a companion site to 6. ppt - Free download as Powerpoint Presentation (. One would be experimental in nature, where we repeatedly conduct an experiment. Probability theory is important to empirical sci-entists because it gives them a rational frame w ork to mak e inferences and test ÐÏ à¡± á> þÿ C E þÿÿÿX Y Z [ D F ý Mar 24, 2019 · This document provides an overview of probability theory, including key definitions, concepts, and calculations. The document discusses basic probability concepts including classical, relative frequency, subjective probability, and properties of probability. 3 experimental roots. Example: Selecting From an Jar of Balls. probability play a starring role. It gives detail description about probability, types of probability, difference between mutually exclusive events and independent events, difference between conditional and unconditional probability and Bayes' theorem. Balaji P. Part of the process of Risk Analysis. Events are collections of outcomes. a) discuss laws of probability, which are useful ; b) define combinations (and permutations) c) use a) and b) to develop the binomial distribution, which is useful. A variable represents an event (a subset of the space of possible outcomes). MIT OpenCourseWare is a web based publication of virtually all MIT course content. If X is a set and Y is a set, and there is a sequence of well-specified operations for assigning a well-defined object to every element , and by applying these sequence of operations to every member of a set X Probability has been introduced in Maths to predict how likely events are to happen. F- Test 8. , coin flips, packet arrivals, noise voltage • Basic elements of probability: Sample space: The set of all possible “elementary” or “finest grain” Jun 23, 2012 · Since there is only one outcome, it is a simple event. (also called the complement of A) 19. 28). probability theory) and the integral theory as a Daniell functional. Read more. Probability is quantified as a number between 0 and 1, where, loosely speaking, 0 indicates impossibility and 1 indicates certainty. Brajesh Kumar Jha Page 36 Theorems on Probability Theorem 1 If A is any event defined on finite sample space U, then ( ) ( )APAP −= 1' Where A’ is the complementary event of A. g. Sc mathematics and statistics students. 2 Complementary events If the event is neither impossible nor certain, then clearly its probability is between 0 and 1. P (A) =1, indicates total certainty in an event A. Probabilities can be expressed at fractions, decimals, or percents. Jun 5, 2017 · Probability 10th class. Sample space = {1,2,3,4,5,6} Event = {}, i. Resource Type: Lecture Notes. Find the probability of each outcome. Review of basic probability theory. This work is in the public domain. Two coins are tossed. pathogen on human health. It introduces key concepts such as random experiments, sample spaces, events, assigning probabilities, conditional probability, independent events, and random variables. To learn probability theory and sample space concepts as they pertain to quantifying uncertaint. There are several ways of viewing probability. 28. 426 views • 22 slides Mar 9, 2015 · If events A and B are independent, the probability of both events occurring is found by multiplying the probabilities of the events. 3 blue. Find the probability that he gets a red and then a blue ball, in that order. All 6 faces of a die: Aug 17, 2023 · Introduction to Statistics is a resource for learning and teaching introductory statistics. Abhishekkumarkushwah7. Events with probability close to one are more likely to occur. ? Trials refers to an event whose outcome is un-known. 1-26; 1-28 (Balasubramanian) 2-2; 2-4 code; 2-9 code; 2-11 code; 2 Jun 27, 2017 · A Probability Distribution is a way to shape the sample data to make predictions and draw conclusions about an entire population. Examples are provided for each concept to illustrate Probability and Statistics for Data Science - Spring 2016 Lecture Slides (1) Probability Theory . 676 kB. characterization. 4. 29). Conditional probability considers the probability of one event occurring given that another event has Use probability theory as a formal means of manipulating degrees of belief Given a proposition, A, assign a probability, P(A), such that 0 = P(A) = 1, where if A is true, P(A)=1, and if A is false, P(A)=0. The laws of chance; 2 overview. The document discusses random experiments, sample spaces Set books The notes cover only material in the Probability I course. The theory of probability has always been associated with gambling and many most accessible examples still come from that activity. txt) or read online for free. . The document defines basic probability concepts such as experiments, sample spaces, simple events, events, complements, and subjective and objective views of probability. Proposition A must be either true or false, but P(A) summarizes our degree of belief in A being true/false. • Place 100 marbles in a bag; 35 blue, 45 red and 20 yellow. It defines probability as a measure of how often an event will occur if an experiment is repeated. AI-enhanced description. An estimate of the probability of occurrence and. Example 1: finding the probability of an event a. 233 kB. A fair coin gives you Heads May 26, 2022 · 13. Moreover, at a certain point, Schwartz’ theory of distributions and discuss elements of the Fourier analysis become useful, and eventually, the reader is referred to our second part-book [96]. Sep 4, 2012 · Probability- General Rules 1. steps hazard identification, exposure. The probability that a drawing pin will land ‘point up’ is 0:62. Chapter Five Elementary Probability Theory. 670 views • 47 slides Mar 24, 2018 · 11. 36. Even More Brownian Motion (PDF) 39. Suppose you’ve already texted the characters “There in a minu” You’d like your mobile phone to guess the most likely completion of “minu” rather than MINUET or MINUS or MINUSCULE. The document defines probability and provides examples of calculating probabilities using tree diagrams and tables. A given probability can take on a value ranging from 0 to 1. Jul 30, 2012 • Download as PPT, PDF •. It provides an example to calculate unconditional, conditional, and joint probabilities using a table of frequency data. u k location Bioscience Building (New Biology), 1 Elementary Probability Theory. 27. assessment, hazard characterization, and risk. P ( ) = 0 3. Documents; view. You've experienced probability when you've flipped a coin, rolled some dice, or looked at a weather forecast. The probability of an event is a number indicating how likely that event will occur. It discusses basic probability concepts like Mar 27, 2018 · Any probability discloses a pattern of behavior that is expected to occur in the long run. The videos in Part I introduce the general framework of probability models, multiple discrete or continuous random variables, expectations, conditional distributions, and various powerful tools of general applicability. A function from I to M is a rule that associates to each element of I a corresponding element of M. mr zm ke hz gt hd ec ly af yc