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CHAPTER
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Probability and Random Variables
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IN THIS CHAPTER Summary: We ve completed the basics of data analysis and we now begin the transition to inference In order to do inference, we need to use the language of probability In order to use the language of probability, we need an understanding of random variables and probabilities The next two chapters lay the probability foundation for inference In this chapter, we ll learn about the basic rules of probability, what it means for events to be independent, and about discrete and continuous random variables, simulation, and rules for combining random variables
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Probability Random Variables Discrete Random Variables Continuous Random Variables Probability Distributions Normal Probability Simulation Transforming and Combining Random Variables
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The second major part of a course in statistics involves making inferences about populations based on sample data (the first was exploratory data analysis) The ability to do this is based on being able to make statements such as, The probability of getting a finding as different, or more different, from expected as we got by chance alone, under the assumption that the
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143
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144 U Step 4 Review the Knowledge You Need to Score High
null hypothesis is true, is 06 To make sense of this statement, you need to have a understanding of what is meant by the term probability as well as an understanding of some of the basics of probability theory An experiment or chance experiment (random phenomenon): An activity whose outcome we can observe or measure but we do not know how it will turn out on any single trial Note that this is a somewhat different meaning of the word experiment than we developed in the last chapter example: if we roll a die, we know that we will get a 1, 2, 3, 4, 5, or 6, but we don t know which one of these we will get on the next trial Assuming a fair die, however, we do have a good idea of approximately what proportion of each possible outcome we will get over a large number of trials Outcome: One of the possible results of an experiment (random phenomenon) example: the possible outcomes for the roll of a single die are 1, 2, 3, 4, 5, 6 Individual outcomes are sometimes called simple events
Sample Spaces and Events
Sample space: The set of all possible outcomes, or simple events, of an experiment example: For the roll of a single die, S = {1, 2, 3, 4, 5, 6} Event: A collection of outcomes or simple events That is, an event is a subset of the sample space example: For the roll of a single die, the sample space (all outcomes or simple events) is S = {1, 2, 3, 4, 5, 6} Let event A = the value of the die is 6 Then A = {6} Let B = the face value is less than 4 Then B = {1, 2, 3} Events A and B are subsets of the sample space example: Consider the experiment of flipping two coins and noting whether each coin lands heads or tails The sample space is S = {HH, HT, TH, TT} Let event B = at least one coin shows a head Then B = {HH, HT, TH} Event B is a subset of the sample space S Probability of an event: the relative frequency of the outcome That is, it is the fraction of time that the outcome would occur if the experiment were repeated indefinitely If we let E = the event in question, s = the number of ways an outcome can succeed, and f = the number of ways an outcome can fail, then P (E) = s s+ f
Note that s + f equals the number of outcomes in the sample space Another way to think of this is that the probability of an event is the sum of the probabilities of all outcomes that make up the event For any event A, P(A) ranges from 0 to 1, inclusive That is, 0 P(A) 1 This is an algebraic result from the definition of probability when success is guaranteed ( f = 0, s = 1) or failure is guaranteed ( f = 1, s = 0) The sum of the probabilities of all possible outcomes in a sample space is one That is, if the sample space is composed of n possible outcomes,
p = 1