unit2 - Stat131A 1 Statistics 131A Spring 2011 Unit 2...

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Unformatted text preview: Stat131A 1 Statistics 131A, Spring 2011 Unit 2: Probability Chris Paciorek Stat131A 2 Outline of Material Introduction to Probability Discrete Random Variables and Probability Distributions Continuous Random Variables and Probability Distributions Joint Probability Distributions and Random Samples Stat131A 3 Sources I MS: Chapters 3-6, excluding 3.8, 3.9, 4.8, 4.9, 4.11, 5.7, 5.8, 5.9, 5.10, most of 6.6, 6.10, 6.11 I Basically we’re excluding some of the more technical mathematics. Stat131A 4 Introduction to Probability Randomness I Randomness means that the possible outcomes are known, but for any given observation, the outcome is uncertain. I Probability quantifies the chance of each possible outcome when the outcomes are random. I Example: I A coin flip: what are the possible outcomes? Is the outcome of a given coin flip random? I Randomly choose a person in the US and find out their age: What are the possible outcomes? Stat131A 5 Introduction to Probability Defining probability I One definition of probability is that the probability of an outcome is the long-run frequency of that outcome. I First, the pattern in the short-run can differ from what you might expect. I Next, let’s look at the long-run frequency of heads in coin-flipping. [see R demo] I Note that in a small sample, the empirical proportion does not equal 0.50 because of the randomness. Stat131A 6 Introduction to Probability Things can get complicated I What is the probability that it will rain tomorrow? Is there any long-run frequency? I What is the probability that the Sierra snowpack at the end of winter will decrease with climate change? Is there a long-run frequency? I What is the probability that an individual has a disease, not having any other information. Is this a random event? I What is the probability that a randomly selected individual in the US has a disease. Is this random? Stat131A 7 Introduction to Probability Subjective probability I The subjective view of probability is that probability summarizes a person’s uncertainty (or degree of belief) about an outcome, based on an evaluation of the information available to the person. I Example: how could we make a probability statement about the Sierra snowpack? I In this version, one way of looking at probability is that it determines how you should bet on an outcome. If your probability is 0.10 (odds of 9:1), you should need to receive at least $9 for winning a bet in order to place a $1 bet on the outcome. I We’ll generally work with the long-run frequency interpretation in this class, but there is an important branch of statistics called Bayesian statistics, which is grounded in a subjective probability interpretation. Stat131A 8 Introduction to Probability The sample space and events I The sample space, S , is the collection of all the possible outcomes (MS calls these ’simple events’)....
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This note was uploaded on 09/11/2011 for the course STAT 131A taught by Professor Isber during the Spring '08 term at Berkeley.

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unit2 - Stat131A 1 Statistics 131A Spring 2011 Unit 2...

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