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Home Page Title Page Contents Page 1 of 37 Go Back Full Screen Close Quit First Prev Next Last Go Back Full Screen Close Quit STAT231: STATISTICS Paul Marriott [email protected] May 8, 2008
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Home Page Title Page Contents Page 2 of 37 Go Back Full Screen Close Quit First Prev Next Last Go Back Full Screen Close Quit Chapter 2: Statistical Science Statistical science, or statistics, is the discipline that deals with the collection, analysis and interpretation of data, and with the study and treatment of variability and of uncertainty. Probability models are used to describe random processes. (For convenience we’ll often use the single term “process” below but the terms population or phenomenon could also be inserted.) They help us understand such processes and to make deci- sions in the face of uncertainty.
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Home Page Title Page Contents Page 3 of 37 Go Back Full Screen Close Quit First Prev Next Last Go Back Full Screen Close Quit Probability models Probability models are used to describe random processes. (i) when studying a process scientifically, questions are often for- mulated in terms of a model for the process, (ii) the data collected in studying processes are variable, so ran- dom variables are often used in discussing and dealing with data, (iii) studies of a process usually lead to inferences or decisions that involve some degree of uncertainty, and probability is used to quantify this. (iv) procedures for making decisions are often formulated in terms of models, and it is necessary to check that the models “agree” with observed data. (v) data are usually needed in the development of a model. In addition, models allow us to characterize processes, and to sim- ulate them via computer experiments or other means.
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Home Page Title Page Contents Page 4 of 37 Go Back Full Screen Close Quit First Prev Next Last Go Back Full Screen Close Quit Probability Distributions Consider a variable Y associated with the units in a popula- tion or process. To describe or “model” the variability in Y -values we use probability distributions, which were introduced in Stat 230. This is done as follows: let Y be the Y -value for a randomly chosen unit in the population or process. Because the value is random we now call Y a random vari- able , and use a probability distribution to provide us with probabilities such as Pr ( a Y b ) . A discrete random variable (r.v.) Y is one for which the range R (set of possible values) of Y is countable. A continuous r.v. is one whole range R consists of one or more continuous intervals of real numbers.
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Home Page Title Page Contents Page 5 of 37 Go Back Full Screen Close Quit First Prev Next Last Go Back Full Screen Close Quit Example: Binomial Distribution Consider a “toy” example in which a six-sided die is rolled repeatedly for a total of n times.
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