Lecture 1: Review of Probability

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1 EAS31136/B9036: Statistics in Earth & Atmospheric Sciences Lecture 1: Review of Probability Instructor: Prof. Johnny Luo

2 Dates Topic Reading (Based on the 2 nd Edition of Wilks book) Other Activity Aug 31 Introduction; Review of probability Wilks, Chap 2 Pre-test Sep 7 Matlab tutorial (optional) Sep 14 Review of probability; Probability Distribution 1 Wilks, Chap 2, 3 Sep 21 Probability Distribution 2 Wilks, Chap 3, 4 Sep 28 Hypothesis testing Wilks, Chap 5 Oct 5 Linear regression I Wilks Chap 6; von Storch 8-9 Oct 12 Linear regression II Wilks Chap 6; von Storch 8-9 Oct 19 Time series analysis I Wilks 8; von Storch Oct 26 Midterm; discussion of final project Nov 2 Time series analysis II Wilks 8; von Storch Nov 9 Principal Component Analysis & Empirical orthogonal functions I Wilks 11; von Storch 13 Project 1-page abstract due Nov 16 Principal Component Analysis & Empirical orthogonal functions II Wilks 11; von Storch 13 Project progress report due Nov 30 Cluster analysis Wilks 14 Dec 7 Final project presentation

3 Outlines 1. Definition of terms 2. Axioms of Probability & different views 3. Some properties of probability

4 Probability deals with uncertainties Ø When facing uncertainties, we need a way to describe it. We can go with qualitative descriptors such as rain likely, unlikely or possible. Ø Probability is a quantitative way of expressing uncertainty, e.g., 40% chance of rain. Ø (Dictionary) Probability: the extent to which an event is likely to occur, measured by the ratio of the favorable cases to the whole number of cases possible. Ø Probability builds upon an abstract mathematical system.

5 A Few Terms Ø Events: A set of possible (uncertain) outcomes (e.g., flipping coin: you won t know for sure which face will come up). Ø Sample Space (or Event Space): the set of all possible events. Usually use capital letter S to represent it. Ø Mutually Exclusive and Collectively Exhaustive (MECE) events; ME: no more than one of the events can occur; CE: at least one of the events will occur.

6 These are null space Venn Diagram

7 4. There are only three companies that make electric cars: companies A, B, and C. Market shares of companies, A, B, C are 10%, 25%, 65%, respectively. If you randomly pick an electric car, what s the probability it was made by A, by B, or by C? P(A) = P(B) = P(c) = 10% 25% 65%

8 5. Now you know the defect rate of new electric cars is 30%, 20%, and 10% for companies A, B, C, respectively. If you randomly pick an electric car and it is defected, what s the probability it was made by A, by B, or by C? 10% x 30% = % x 20% = % x 10% = % 25% 65%

9 5. Now you know the defect rate of new electric cars is 30%, 20%, and 10% for companies A, B, C, respectively. If you randomly pick an electric car and it is defected, what s the probability it was made by A, by B, or by C? 10% x 30% = % x 20% = % x 10% =0.065 Total = = P(A) = 0.03/0.145 = 21% P(B) = 0.05/0.145 = 34% P(C) = 0.065/0.145 = 45%

10 Outlines 1. Definition of terms 2. Axioms of Probability & different views 3. Some properties of probability

11 Axioms of Probability (Dictionary) Axiom: A self-evident truth that requires no proof; a universally accepted principle; (mathematics) a proposition that is assumed without proof for the sake of studying the consequences that follow from it. For an event E in a sample space S 1.0 P(E) 1 2. P( S) = 1 3. P( E1 È E2) = P( E1) + P( E2) where E1 and E 2 mutually exclusive

12 The axioms are like the US Constitution. They are not very informative about how to estimate and interpret probability. There are two dominant views of the meaning of probability: the Frequency view and the Bayesian view. Frequency view: The true probability of of event {E} exists and can be estimated through a long series of trials. Bayesian view: There is no such a thing as true probability; we just estimate it based on whatever information we have in hand.

13 Monty Hall Problem What s probability of winning for each door? You pick one and then the host will open up one with a goat You are now offered an opportunity to switch your choice. Will switch change the probability for the two remaining doors? What are the winning probabilities for door 1 and door 2 now?

14 Monty Hall Problem What about 100 doors? You pick one. Then, I open 98: all goats! Will you now switch to the remaining one? You are now offered an opportunity to stay with your choice, or switch. Will switch give you better chance?

15 Monty Hall Problem Frequency view: The true prob of of event {E} exists and can be estimated through a long series of trials. You are now offered an opportunity to stay with your choice, or switch. Will switch give you better chance? Bayesian view: There is no such a thing as true prob; we just estimate it based on whatever info we have in hand.

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