By William Feller
Significant adjustments during this variation contain the substitution of probabilistic arguments for combinatorial artifices, and the addition of latest sections on branching procedures, Markov chains, and the De Moivre-Laplace theorem.
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Additional resources for An Introduction to Probability Theory and Its Applications, Volume 2
1 Normality and the Box–Cox transformation The normal distribution plays an important role in many statistical analyses, and nice theoretical results follow if we can assume normality and equal variance, neither of which hold in our example. However, simple power transformations often improve the situation considerably. The Box–Cox transformation (Box and Cox, 1964) is a scale- and location-shifted version of the power transformation, given by fλ (x) = xλ − 1 λ for λ = 0, with f0 (x) = log x. This formulation has the advantage of being continuous with respect to the “power” λ at λ = 0.
To use the data in a lattice plot, it is convenient to coerce it into a data frame. 8. 9. Survival among diﬀerent subgroups of passengers on the Titanic, with a diﬀerent horizontal scale in each panel. This emphasizes the proportion of survivors within each subgroup, rather than the absolute numbers. The proportion of survivors is smallest among third-class passengers, although the absolute number of survivors is not too low compared to the other classes. All this plot really tells us is that there were many more males than females aboard (particularly among the crew, which is the largest group), and that there were even fewer children; which, although true, is unremarkable.
As we saw in Chapter 1, a powerful construct that forces direct comparison is superposition, where data associated with different levels of a grouping variable are rendered together within a panel, but with diﬀerent graphical characteristics. For example, diﬀerent curves could be drawn in diﬀerent color or line type, or points could be drawn with diﬀerent symbols. Superposition is usually more eﬀective than multipanel conditioning when the number of levels of the grouping variable is small. For many lattice functions, specifying a groups argument that refers to a categorical variable is enough to produce a “natural” grouped display.