251probabl 9/28/04 (Open this document in 'Outline' view!)H. Introduction to Probability1. Experiments and Probabilitya. Definition and rules for Statistical Probability.b. An Event.c. Symmetrical, Statistical and Subjective Probability.2. The Venn Diagram.a. The Addition Rule. .(i) Meaning of Union (or).(ii) Meaning of Intersection (and).b. Meaning of Complement. .c. Extended Addition Rule.3. Conditional and Joint Probability.a. The Multiplication Rule orb. A Joint Probability Table.c. Extended Multiplication Rule.d. Bayes' Rule.4. Statistical Independence.a. Definition:b. Consequence:c. Consequence: If and are independent so are etc.5. Review.I. Permutations and Combinations.1. Counting Rule for Outcomes.a. If an experiment has steps and there arepossible outcomes on the first step, possibleoutcomes on the second step, etc. up to possibleoutcomes on the th step, then the total number ofpossible outcomes is the product .b. Consequence. If there are exactly outcomes ateach step, the total possible outcomes from steps is.2. Permutations.a. The number of ways that one can arrange nobjects:b. Order counts!3. Combinations.a. Order doesn't count!b. Probability of getting a given combinationJ. Random Variables.1. Definitions.2. Probability Distribution of a Discrete RandomVariable.3. Expected Value (Expectation) of a Discrete Random Variable..Rules for linear functions of a random variable:a.b.c.d.4. Variance of a Discrete Random Variable.Rules for linear functions of a random Variable:a.b.c.d.5. Summarya. Rules for Means and Variances of Functions ofRandom Variables. 251probex4b. Standardized Random Variables, .See 251probex2.6. Continuous Random Variables.a. Normal Distribution (Overview).b. The Continuous Uniform Distribution.c. Cumulative Distributions, Means and Variancesfor Continuous Distributions.d. Chebyshef's Inequality Again.7. Skewness and Kurtosis (Short Summary).251probabl 9/28/04 (Open this document in 'Outline' view!)H. Introduction to Probability1. Experiments and ProbabilityDefine a random experiment, a sample space, an outcome, a basic outcomea. Definition and rules for Statistical Probability.(i) If 1A is impossible, 01AP.(ii) If 1A is certain, 11AP.(iii) For any Outcome 1A , 101 AP.(iv). If NAAAA ,,,,321 represent all possible outcomes and are mutually exclusive, then 1321NAPAPAPAP.b. An Event.c. Symmetrical, Statistical and Subjective Probability.2. The Venn Diagram.A diagram representing events as sets of points or ‘puddles.’a. The Addition Rule. BAPBPAPBAP .(i) Meaning of Union (or). BAP means the probability of A occurring or ofB occurring or both. It always includes BAP if it exists.(ii) Meaning of Intersection (and). BAP means the probability of both A and B occurring. Note that if A and B are mutually exclusive 0BAP Diagram for dice problems.------------------------------------654321654321b. Meaning of Complement. APAP 1.This event can be called '' Anot. Note that if A and B are collectively exhaustive 1BAP . If A and B are both collectively exhaustive and mutually exclusive B is the complement of A.c. Extended Addition Rule. CBAP CBAPCBPCAPBAPCPBPAP 3. Conditional and Joint Probability.a. The Multiplication Rule BPBAPBAP or BPBAPBAP.The conditional probability of A given B, BPBAPBAPis the probability of event A assuming that event B has occurred.2b. A Joint Probability Table.What is the difference between joint, marginal and conditional probabilities? Remember that we cannot read a conditional probability directly from a joint probability table but must compute it using the second version of the Multiplication Rule.c. Extended Multiplication Rule. APABPBACPCBAP d. Bayes' Rule. APBPBAPABP 4. Statistical Independence.a. Definition: APBAP b. Consequence: BPAPBAP c. Consequence: If A and B are independent so are BABA and , and etc.5. Review.Rule In GeneralAandBmutually exclusiveAandBindependentMultiplication BAP APABPBPBAP0 BPAPAddition BAP BAPBPAP BPAP BPAPBPAPBayes' Rule BAP BPAPABP0 APBayes' Rule ABP APBPBAP0 BPI. Permutations and Combinations.1. Counting Rule for Outcomes.a. If an experiment has k steps and there are 1n possible outcomes on the first step, 2n possible outcomes on the second step, etc. up to nk possible outcomes on the kth step, then the total number of possible outcomes is the product knnn 21.b. Consequence. If there are exactly n outcomes at each step, the total possible outcomes from k steps is kn.2. Permutations.a. The number of ways that one can arrange n objects: !nb. !!rnnPnr Order counts!33. Combinations.a. !!!rrnnCnr Order doesn't count!b. Probability of getting a given combinationThis is the number of ways of getting the specified combination divided by the total number of possible combinations.If there are a equally likely ways to get what you want and b equally likely possible outcomes, the probability of getting the outcomes you want is .ba Example: If there is only one way to get 4 jacks from 4 jacks in a poker hand and 481C waysto get another card, 4811 Ca The number of ways to get a poker hand of 5 cards is ,525Cb so the probability of getting a poker hand with 4 jacks is 525481CCba.J. Random Variables.1. Definitions.Discrete and Continuous Random Variables. Finite and infinite populations. Sampling with replacement.2. Probability Distribution of a Discrete Random Variable.By this we mean either a table with each possible value of a random variable and the probability of each value (These probabilities better add to one!) or a formula that will give us these results. We can still speak of Relative Frequency and
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