Introduction Probability. Math 141. Introduction to Probability and Statistics. Albyn Jones

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1 Math 141 to and Statistics Albyn Jones Mathematics Department Library jones/courses/141 September 3, 2014

2 Motivation How likely is an eruption at Mount Rainier in the next 25 years?

3 Data! Post ice-age eruptions Mount Rainier vs a Poisson Point Process Poisson Mount Rainier Year

4 Two Models Don t worry about the details! Poisson Process: Events uniformly distributed in time. Roughly 50 events in the last years: one every 240 years.

5 Two Models Don t worry about the details! Poisson Process: Events uniformly distributed in time. Roughly 50 events in the last years: one every 240 years. Prediction: roughly a 10% chance of an eruption in the next 25 years, regardless of the elapsed time since the last eruption.

6 Two Models Don t worry about the details! Poisson Process: Events uniformly distributed in time. Roughly 50 events in the last years: one every 240 years. Prediction: roughly a 10% chance of an eruption in the next 25 years, regardless of the elapsed time since the last eruption. Hidden Markov Model: Two (unobservable) states with different rates. Given the last eruption was roughly 1050 years ago, we think we are in a low rate regime: roughly one eruption every 650 years.

7 Two Models Don t worry about the details! Poisson Process: Events uniformly distributed in time. Roughly 50 events in the last years: one every 240 years. Prediction: roughly a 10% chance of an eruption in the next 25 years, regardless of the elapsed time since the last eruption. Hidden Markov Model: Two (unobservable) states with different rates. Given the last eruption was roughly 1050 years ago, we think we are in a low rate regime: roughly one eruption every 650 years. Prediction: roughly a 3.7% chance of an eruption in the next 25 years.

8 Questions Hint: statistical analysis! Which of those predictions is more reliable? In other words: which is the better model?

9 Questions Hint: statistical analysis! Which of those predictions is more reliable? In other words: which is the better model? How do we produce estimates for those models?

10 Questions Hint: statistical analysis! Which of those predictions is more reliable? In other words: which is the better model? How do we produce estimates for those models? How accurate or trustworthy are those estimates?

11 Questions Hint: statistical analysis! Which of those predictions is more reliable? In other words: which is the better model? How do we produce estimates for those models? How accurate or trustworthy are those estimates? How do we validate the models?

12 Statistics what is it all about? Formal inference: estimates, confidence intervals and hypothesis tests; quantification of uncertainty.

13 Statistics what is it all about? Formal inference: estimates, confidence intervals and hypothesis tests; quantification of uncertainty. Tools: probability theory, computational engines like R.

14 Statistics what is it all about? Formal inference: estimates, confidence intervals and hypothesis tests; quantification of uncertainty. Tools: probability theory, computational engines like R. Informal inference: judgements about statistical models model choice, model validation

15 Statistics what is it all about? Formal inference: estimates, confidence intervals and hypothesis tests; quantification of uncertainty. Tools: probability theory, computational engines like R. Informal inference: judgements about statistical models model choice, model validation Tools: graphical methods, computational engines like R.

16 Statistics what is it all about? Formal inference: estimates, confidence intervals and hypothesis tests; quantification of uncertainty. Tools: probability theory, computational engines like R. Informal inference: judgements about statistical models model choice, model validation Tools: graphical methods, computational engines like R. Note the computational theme!

17 A Little Theory The mathematics we need to quantify uncertainty A little History: gambling! dice! cards! A little Philosophy: epistemology and subjective probability, positivism and objective probability.

18 Example Toss a fair coin 3 times. What is the probability that two of the three tosses yield heads? We need some terminology and notation: Sample Space: the set of possible outcomes.

19 Example Toss a fair coin 3 times. What is the probability that two of the three tosses yield heads? We need some terminology and notation: Sample Space: the set of possible outcomes. Event: a subset of the sample space.

20 Example Toss a fair coin 3 times. What is the probability that two of the three tosses yield heads? We need some terminology and notation: Sample Space: the set of possible outcomes. Event: a subset of the sample space. : a function assigning real numbers to events.

21 Sample Space: Ω Toss a fair coin 3 times. What are the possible outcomes? {HHH} {HHT }, {HTH}, {THH} {HTT }, {THT }, {TTH} {TTT } These are the events in our sample space.

22 Notation A little set theory Let A and B be events (subsets of the sample space Ω). Term Notation Interpretation Union A B A or B occurs (or both!) Intersection A B A and B both occur Complement A c,!a, (Ω \ A) A does not occur Disjoint Events A B = A and B can not both occur

23 Notation: Examples three coin tosses again two heads : a union of three events {HHT } {HTH} {THH}

24 Notation: Examples three coin tosses again two heads : a union of three events {HHT } {HTH} {THH} at least one head : the complement of no heads {TTT } c = {TTH} {THT }... {HHH}

25 Notation: Examples three coin tosses again two heads : a union of three events {HHT } {HTH} {THH} at least one head : the complement of no heads {TTT } c = {TTH} {THT }... {HHH} an impossible event! {TTT } {HHH}

26 three coin tosses again Let s assign probabilities to our 8 events, giving each event in Ω the same probability (why?). P{HHH} = P{HHT } =... P{TTT } = 1 8 Note the probabilities of the 8 events add up to 1.

27 More! three coin tosses again Now, what is the probability of getting two heads in three tosses? P{HHT } = P{HTH} = P{THH} = 1 8 The probabilities of these 3 events add up to 3/8. Is that the correct value for the probability of getting two heads? We need some rules for computing probabilities!

28 Rules for aka AXIOMS Let Ω be a sample space, and E 1, E 2, E 3,... be events. P : {Events} R according to the following three rules:

29 Rules for aka AXIOMS Let Ω be a sample space, and E 1, E 2, E 3,... be events. P : {Events} R according to the following three rules: 1 For any event E: 0 P(E) 1

30 Rules for aka AXIOMS Let Ω be a sample space, and E 1, E 2, E 3,... be events. P : {Events} R according to the following three rules: 1 For any event E: 2 P(Ω) = 1 0 P(E) 1

31 Rules for aka AXIOMS Let Ω be a sample space, and E 1, E 2, E 3,... be events. P : {Events} R according to the following three rules: 1 For any event E: 2 P(Ω) = 1 0 P(E) 1 3 If E 1, E 2, E 3,... are disjoint events, then P(E 1 E 2...) = P(E i ) = P(E 1 ) + P(E 2 ) +...

32 Example three coin tosses again Now, what is the probability of getting two heads in three tosses, given our assignment of equal probability to each: P({HHT }) = P({HTH}) = P({THH}) = 1 8

33 Example three coin tosses again Now, what is the probability of getting two heads in three tosses, given our assignment of equal probability to each: P({HHT }) = P({HTH}) = P({THH}) = 1 8 Are these events disjoint?

34 Example three coin tosses again Now, what is the probability of getting two heads in three tosses, given our assignment of equal probability to each: P({HHT }) = P({HTH}) = P({THH}) = 1 8 Are these events disjoint? yes!

35 Example three coin tosses again Now, what is the probability of getting two heads in three tosses, given our assignment of equal probability to each: P({HHT }) = P({HTH}) = P({THH}) = 1 8 Are these events disjoint? yes! Therefore, by axiom 3, P({HHT } {HTH} {THH}) = P({HHT }) + P({HTH}) + P({THH}) = 3 8

36 Complementary Events What do we know about the events E and E c? What is E E c? What is E E c?

37 More on Complementary Events For any event E, E E c =, so E and E c are disjoint. For any event E, E E c = Ω. Putting these facts together with our axioms: 1 = P(Ω) = P(E E c ) = P(E) + P(E c ) Thus P(E c ) = 1 P(E)

38 Example: Complementary Events What is the probability of at least one head in three tosses?

39 Example: Complementary Events What is the probability of at least one head in three tosses? What is the complement of at least one head?

40 Example: Complementary Events What is the probability of at least one head in three tosses? What is the complement of at least one head? No heads! (All tails.)

41 Example: Complementary Events What is the probability of at least one head in three tosses? What is the complement of at least one head? No heads! (All tails.) Using the last result we have P(at least one head) = P({TTT } c ) = 1 P({TTT }) = = 7 8

42 A probability inequality If A B, then P(A) P(B), proof by picture: A B

43 A General Addition Formula: Inclusion/Exclusion P(A B) = P(A) + P(B) P(A B) A B

44 Summary 1 Definitions: Sample Space, Events, Disjoint Events 2 Axioms or Rules of 3 P(E) = 1 P(E c ) 4 Addition formula: P(A B) = P(A) + P(B) P(A B)

45 Assignment! Read Chapter 2.

Events A and B are said to be independent if the occurrence of A does not affect the probability of B.

Independent Events Events A and B are said to be independent if the occurrence of A does not affect the probability of B. Probability experiment of flipping a coin and rolling a dice. Sample Space: {(H,