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Probability and Sample Analysis

This lecture discusses the importance of probability in analyzing samples and estimating properties of populations. It covers topics such as experiments, sample space, and events in probability theory.

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Probability and Sample Analysis

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  1. 45-733: lecture 2 (chapter 3) Probability William B. Vogt, Carnegie Mellon, 45-733

  2. What do we do with statistics? • Describe: a sample well • Analyze: the sample to estimate properties of the population • Analyze: the analysis to describe how sure we are of it William B. Vogt, Carnegie Mellon, 45-733

  3. Why do we need probability? • Utility outside statistics • Gambling, physics, chemistry, asset pricing, insurance, etc William B. Vogt, Carnegie Mellon, 45-733

  4. Why do we need probability? • Utility within statistics • When we are describing how sure we are that our analysis of the population is right, probability gives us a precise language in which to speak. • We will want to say things like: • I am more than 95% sure that US household income is greater than $30,000 • I am 99% certain that the mean time to failure of our light bulbs is between 100 and 120 hours • I am 80% sure that GDP growth will be between 1.2% and 3.5% next year William B. Vogt, Carnegie Mellon, 45-733

  5. Experiment, sample space, event • Consider an experiment • A process which can have one of several possible outcomes • Which outcome will occur is unknown to the experimenter or observer • Examples • Coin toss, die throw • Light bulb tested to failure • Economy evolves for one year William B. Vogt, Carnegie Mellon, 45-733

  6. Experiment, sample space, event • The sample space • A list of all the possible outcomes of an experiment • Examples • Coin toss: sample space = [heads,tails] • Die throw: sample space=[1,2,3,4,5,6] • Light bulb failure time: S=[all positive real numbers] • Economy growth: S=[all real numbers > -100%] William B. Vogt, Carnegie Mellon, 45-733

  7. Experiment, sample space, event • A basicevent • One point in the sample space • Examples • Coin toss: heads • Die throw: 3 • Light bulb: 400 hours • Economy growth: 1.7% William B. Vogt, Carnegie Mellon, 45-733

  8. Experiment, sample space, event • An event • A collection of one or more basic events • A collection of one or more points in sample space • Examples • Coin toss: “heads” “tails” “heads,tails” • Die throw: “3” “3,6” “1,2,3,4,5,6” • Light bulb: “400 hours” “between 5 and 18 hours” • Economy growth: “1.7%” “2.1%,between –1 and 1%” William B. Vogt, Carnegie Mellon, 45-733

  9. Experiment, sample space, event • Notation • We often write the sample space as S • We often denote basic events as s • We often write events as A, B, C, etc William B. Vogt, Carnegie Mellon, 45-733

  10. Experiment, sample space, event • Venn diagram • A Venn diagram is a way of representing sample space, events, and operations • Elements of Venn diagram • Large rectangle representing the sample space • Circles or other shapes representing events • (optional) points representing basic events William B. Vogt, Carnegie Mellon, 45-733

  11. Experiment, sample space, event • Venn diagram • Example: the sample space of the die throw 1 4 6 3 5 2 William B. Vogt, Carnegie Mellon, 45-733

  12. Experiment, sample space, event B A • Venn diagram • Example: • A=“4,5,3” • B=“3,2,6” 1 4 6 3 5 2 William B. Vogt, Carnegie Mellon, 45-733

  13. Experiment, sample space, event • Notice • All basic events are events • The sample space is an event • There is a special event called the null set or the null event or the empty event. It is =[] William B. Vogt, Carnegie Mellon, 45-733

  14. Experiment, sample space, event • Membership • A basic event may either belong to an event or not • We will write sA when the basic event s is in the event A • We will write sA when the basic event s is not in the event A William B. Vogt, Carnegie Mellon, 45-733

  15. Experiment, sample space, event • Membership • Examples • heads “heads,tails” • 1  “1,4,5” • 1  “3,6” • 1%  “between 0 and 3%” • 140 hours  “between 50 and 100 hours” William B. Vogt, Carnegie Mellon, 45-733

  16. Experiment, sample space, event • Membership: Venn diagram • Example: • A=“4,5,3” • 3  A • 1  A 1 4 6 3 5 2 A William B. Vogt, Carnegie Mellon, 45-733

  17. Experiment, sample space, event • Sub-event (subset) • We say that an event B is a sub-event of A if every member of B is also in A, and we write BA • Examples • “heads”  “heads,tails” • “3,4,5”  “1,2,3,4,5” • “between 1% and 1.3%”  “between 0.5% and 4%” • “3,4,5”  “1,2,3,4” William B. Vogt, Carnegie Mellon, 45-733

  18. Experiment, sample space, event B • Sub-event: Venn diagram • Example: • A=“4,5,3” • B=“4,5” • B  A 1 4 6 3 5 2 A William B. Vogt, Carnegie Mellon, 45-733

  19. Experiment, sample space, event • Intersection • A way of making a new event from two events • The intersection of events A and B is the event consisting of all the basic events A and B have in common. • C=AB means C is the intersection of A and B William B. Vogt, Carnegie Mellon, 45-733

  20. Experiment, sample space, event • Intersection • Examples • “1” = “1,2,3”  “1,4” • “between 1% and 1.5%” =“btw 1% and 2%”  “btw 0.8% and 1.5%” • =“1”  “3,4,5” William B. Vogt, Carnegie Mellon, 45-733

  21. Experiment, sample space, event • Intersection: Venn diagram • Example Intersection: • A=“4,5,3” • B=“3,2,6” • C=A  B=“3” C 1 4 6 5 3 2 William B. Vogt, Carnegie Mellon, 45-733

  22. Experiment, sample space, event • Union • A way of making a new event from two events • The union of two events is the event which contains all the basic events which are in either. • C=AB says C is the union of A and B --- C contains all the basic events in either A or B William B. Vogt, Carnegie Mellon, 45-733

  23. Experiment, sample space, event • Union • Examples: • “1,2,3” = “1,2”  “2,3” • “btw 1% and 3%” = “btw 1% and 1.5%”  “btw 1.1% and 3%” • “heads” = “heads”   William B. Vogt, Carnegie Mellon, 45-733

  24. Experiment, sample space, event • Union: Venn diagram • Example Union: • A=“4,5,3” • B=“3,2,6” • D=A  B =“2,3,4,5,6” D 1 4 6 3 5 2 William B. Vogt, Carnegie Mellon, 45-733

  25. Experiment, sample space, event • Mutual Exclusivity • A and B share no basic events in common • A B= • Example: A=“1,4” B=“3,2” 1 4 6 B 3 5 A 2 William B. Vogt, Carnegie Mellon, 45-733

  26. Experiment, sample space, event • Collective exhaustivity • A bunch of events are collectively exhaustive if their union is the sample space • Example: E1=“1,4” E2=“3,2,6” E3=“3,4,5” • E1  E2 E3=“1,2,3,4,5,6” William B. Vogt, Carnegie Mellon, 45-733

  27. Experiment, sample space, event • Collective exhaustivity • A bunch of events are collectively exhaustive if their union is the sample space 1 4 6 3 5 2 William B. Vogt, Carnegie Mellon, 45-733

  28. Experiment, sample space, event • Partitioning • A bunch of events partition the sample space if they are mutually exclusive and collectively exhaustive 1 4 6 3 5 2 William B. Vogt, Carnegie Mellon, 45-733

  29. Experiment, sample space, event • Complement • A complement is all the basic events in the sample space which are not in A • Complements are partitioning 1 4 6 3 5 2 William B. Vogt, Carnegie Mellon, 45-733

  30. Experiment, sample space, event • Some useful rules 1 4 6 3 5 2 William B. Vogt, Carnegie Mellon, 45-733

  31. Experiment, sample space, event • Some useful rules William B. Vogt, Carnegie Mellon, 45-733

  32. Experiment, sample space, event • Some useful rules • If E1,E2,E3,…,Ek are partition the sample space, then • E1 A, E2 A, E3 A,…,Ek A, are mutually exclusive • (E1 A) (E2 A) (E3 A)  … (Ek A)=A William B. Vogt, Carnegie Mellon, 45-733

  33. Experiment, sample space, event • Some useful rules William B. Vogt, Carnegie Mellon, 45-733

  34. What is probability? • Probability is a language within which to describe uncertainty • Uncertainty about which event will occur • Some events are more likely than others • When one event occurs, that may make other events more/less likely to occur • Since it is a language it has rules • Since it is a mathematical language, the rules are precise • Since the rules are precise, the statements it can make are correspondingly precise William B. Vogt, Carnegie Mellon, 45-733

  35. What is probability? • Probability is a number between 0 and 1 • When we say the probability of an event is 0, that means it is impossible for the event to occur • When we say the probability of an event is 1, that means it is certain that the event will occur • If the probability of A occurring is greater than the probability of B occurring, that means that A is more likely than B William B. Vogt, Carnegie Mellon, 45-733

  36. What is probability? • There are differing interpretations of what this number between 0 and 1 means (in terms of the external world) • Frequentist • Imagine doing an experiment many independent times • Each time, we record whether or not the event A occurred • As N (number of experiments) goes to infinity • P(A) = NA/N William B. Vogt, Carnegie Mellon, 45-733

  37. What is probability? • There are differing interpretations of what this number between 0 and 1 means (in terms of the external world) • Subjectivist • The probability of an event A occurring exists only in our minds, reflecting our uncertainty/ignorance • When I say P(A)=0.5 that means I think a 1:1 bet on whether A occurs is a fair bet • When I say P(A)=0.33 that means I think a 2:1 bet on whether A occurs is a fair bet William B. Vogt, Carnegie Mellon, 45-733

  38. What is probability? • Venn diagram • It is often useful to think of probability as area in a Venn diagram A B William B. Vogt, Carnegie Mellon, 45-733

  39. Postulates of probability • For any event A, 0P(A) 1 • For any event A, P(A)=sAP(s) • The probability of an event is just the sum of the probabilities of the basic events which make it up • P(“1,2,3”)=P(1)+P(2)+P(3) • P(S)=1 and P()=0 William B. Vogt, Carnegie Mellon, 45-733

  40. Consequences • If S has n equally likely basic events, each one has probability 1/n • If S has n equally likely basic events and nA of them are in A, then A has probability nA/n William B. Vogt, Carnegie Mellon, 45-733

  41. Consequences • If A and B are mutually exclusive events, then P(A B)=P(A)+P(B) 1 4 6 A 3 B 5 2 William B. Vogt, Carnegie Mellon, 45-733

  42. Consequences • In general P(A B)P(A)+P(B) 1 4 6 A 3 B 5 2 William B. Vogt, Carnegie Mellon, 45-733

  43. Consequences • If B  A, then P(B) P(A) B A William B. Vogt, Carnegie Mellon, 45-733

  44. Rules of probability William B. Vogt, Carnegie Mellon, 45-733

  45. Rules of probability A B William B. Vogt, Carnegie Mellon, 45-733

  46. Conditional probability • Conditional probability is used to deal with partial information • Suppose there are two events, A and B and we wish to know the probability of A occurring • Estimate this probability somehow • Now we learn that B has occurred • How should we change our assessment of the probability of A occurring given that we know for sure B has occurred? William B. Vogt, Carnegie Mellon, 45-733

  47. Conditional probability • Example, die throw • A=“1,2,3,5” • B=“3,4” • P(A)=1/6+1/6+1/6+1/6=2/3 • The probability of A given that we know B has occurred is ½ • Two equally likely outcomes in B • Only one of them is also in A William B. Vogt, Carnegie Mellon, 45-733

  48. Conditional probability • Notation • We write P(A|B) • This is said “The probability of A given B”or “The probability of A conditional on B” • So, we could write P(A|B)=1/2 in our previous example William B. Vogt, Carnegie Mellon, 45-733

  49. Conditional probability • Rule for calculating A B William B. Vogt, Carnegie Mellon, 45-733

  50. Conditional probability • Rule for calculating • In die throw, A=“1,2,3,5” B=“3,4” William B. Vogt, Carnegie Mellon, 45-733

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