Titlebook: All of Statistics; A Concise Course in Larry Wasserman Textbook 2004 Springer Science+Business Media, LLC, part of Springer Nature 2004 Bo

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ProbabilityProbability is a mathematical language for quantifying uncertainty. In this Chapter we introduce the basic concepts underlying probability theory. We begin with the sample space, which is the set of possible outcomes.

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InequalitiesInequalities are useful for bounding quantities that might otherwise be hard to compute. They will also be used in the theory of convergence which is discussed in the next chapter. Our first inequality is Markov’s inequality.

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Models, Statistical Inference and LearningStatistical inference, or “learning” as it is called in computer science, is the process of using data to infer the distribution that generated the data. A typical statistical inference question is:

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Estimating the CDF and Statistical FunctionalsThe first inference problem we will consider is nonparametric estimation of the CDF .. Then we will estimate statistical functionals, which are functions of CDF, such as the mean, the variance, and the correlation. The nonparametric method for estimating functionals is called the plug-in method.

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The BootstrapThe . is a method for estimating standard errors and computing confidence intervals. Let .. = .(..,…, ..) be a ., that is, .. is any function of the data. Suppose we want to know V.(..), the variance of ... We have written V. to emphasize that the variance usually depends on the unknown distribution function ..

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Hypothesis Testing and p-valuesSuppose we want to know if exposure to asbestos is associated with lung disease. We take some rats and randomly divide them into two groups. We expose one group to asbestos and leave the second group unexposed. Then we compare the disease rate in the two groups. Consider the following two hypotheses: