By William Feller
Significant alterations during this version contain the substitution of probabilistic arguments for combinatorial artifices, and the addition of latest sections on branching strategies, Markov chains, and the De Moivre-Laplace theorem.
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Additional info for An Introduction to Probability Theory and Its Applications, Volume 2
Consider this expression: 2 1 3142 If you add first, you will evaluate the expression as 5 1 4 2 5 20 but if you multiply first, the expression becomes 2 1 12 5 14 Obviously, it is crucial to complete the steps of a calculation in the correct order. The basic rules of precedence are to find all squares and square roots first, then do all multiplication and division, and finally complete all addition and subtraction. Thus, the expression 8 1 2 3 22 /2 would be evaluated as 8123 4 8 5 8 1 5 8 1 4 5 12 2 2 The rules of precedence may be overridden by parentheses.
At the ordinal level, scores can be ranked from high to low. At the interval-ratio level, all mathematical operations are permitted. GLOSSARY Data. Information expressed as numbers. Data reduction. Summarizing many scores with a few statistics. Dependent variable. A variable that is identiﬁed as an effect or outcome. The dependent variable is thought to be caused by the independent variable. Descriptive statistics. The branch of statistics concerned with (1) summarizing the distribution of a single variable or (2) measuring the relationship between two or more variables.
First, knowing the level of measurement of a variable is crucial because it tells us which statistics are appropriate and useful. Not all statistics can be used with all variables. 2, different statistics require different mathematical operations. For example, computing an average requires addition and division, and ﬁnding a median (or middle score) requires that the scores be ranked from high to low. Addition and division are appropriate only for intervalratio-level variables, and ranking is possible only for variables that are at least ordinal in level of measurement.
An Introduction to Probability Theory and Its Applications, Volume 2 by William Feller