Standard Probability and Statistics Tables: Student Edition Kokoska and Zwillinger

Table of Contents

Introduction(page 1)

1. Data sets(page 1)

Summarizing Data(page 3)

1. Tabular and graphical procedures(page 3)
   1. Stem-and-leaf plot(page 3)
   2. Frequency distribution(page 3)
   3. Histogram(page 4)
   4. Frequency polygons(page 5)
2. Numerical summary measures(page 5)
   1. (Arithmetic) mean(page 6)
   2. Weighted (arithmetic) mean(page 6)
   3. Geometric mean(page 6)
   4. Harmonic mean(page 7)
   5. Mode(page 7)
   6. Median(page 7)
   7. p% trimmed mean(page 8)
   8. Quartiles(page 8)
   9. Deciles(page 9)
   10. Percentiles(page 9)
   11. Mean deviation(page 9)
   12. Variance(page 10)
   13. Standard deviation(page 10)
   14. Standard errors(page 11)
       1. Standard error of the mean(page 11)
   15. Root mean square(page 11)
   16. Range(page 11)
   17. Interquartile range(page 11)
   18. Quartile deviation(page 11)
   19. Box plots(page 11)
   20. Coefficient of variation(page 14)
   21. Coefficient of quartile variation(page 14)
   22. Z score(page 14)
   23. Moments(page 14)
   24. Measures of skewness(page 15)
       1. Coefficient of skewness(page 15)
       2. Coefficient of momental skewness(page 15)
       3. Pearson's first coefficient of skewness(page 15)
       4. Pearson's second moment of skewness(page 15)
       5. Quartile coefficient of skewness(page 15)
   25. Measures of kurtosis(page 16)
       1. Coefficient of kurtosis(page 16)
       2. Coefficient of excess kurtosis(page 16)
   26. Data transformations(page 16)
   27. Sheppard's corrections for grouping(page 16)

Probability(page 23)

1. Algebra of sets(page 23)
2. Combinatorial methods(page 23)
   1. The product rule for ordered pairs(page 23)
   2. The generalized product rule for k-tuples(page 24)
   3. Permutations(page 24)
   4. Circular permutations(page 24)
   5. Combinations (binomial coefficients)(page 24)
   6. Sample selection(page 25)
   7. Balls into cells(page 25)
   8. Multinomial coefficients(page 26)
   9. Arrangements and derangements(page 27)
3. Probability(page 27)
   1. Relative frequency concept of probability(page 27)
   2. Axioms of probability (discrete sample space)(page 28)
   3. The probability of an event(page 28)
   4. Probability theorems(page 28)
   5. Probability and odds(page 29)
   6. Conditional probability(page 29)
   7. The multiplication rule(page 30)
   8. The law of total probability(page 30)
   9. Bayes' theorem(page 30)
   10. Independence(page 31)
4. Random variables(page 31)
   1. Discrete random variables(page 31)
      1. Probability mass function(page 31)
      2. Cumulative distribution function(page 31)
   2. Continuous random variables(page 32)
      1. Probability density function(page 32)
      2. Cumulative distribution function(page 32)
   3. Random functions(page 32)
5. Mathematical expectation(page 33)
   1. Expected value(page 33)
   2. Variance(page 33)
      1. Theorems(page 33)
   3. Moments(page 34)
      1. Moments about the origin(page 34)
      2. Moments about the mean(page 34)
      3. Factorial moments(page 34)
   4. Generating functions(page 34)
      1. Moment generating function(page 34)
      2. Factorial moment generating functions(page 35)
      3. Factorial moment generating function theorems(page 35)
      4. Cumulant generating function(page 36)
      5. Characteristic function(page 36)
6. Multivariate distributions(page 36)
   1. Discrete case(page 36)
   2. Continuous case(page 37)
   3. Expectation(page 37)
   4. Moments(page 38)
   5. Marginal distributions(page 38)
   6. Independent random variables(page 38)
   7. Conditional distributions(page 39)
   8. Variance and covariance(page 40)
   9. Correlation coefficient(page 40)
   10. Moment generating function(page 40)
   11. Linear combination of random variables(page 41)
7. Inequalities(page 42)

Functions of Random Variables(page 43)

1. Finding the probability distribution(page 43)
   1. Method of distribution functions(page 43)
   2. Method of transformations (one variable)(page 44)
   3. Method of transformations (many variables)(page 45)
   4. Method of moment generating functions(page 47)
2. Sums of random variables(page 47)
   1. Deterministic sums of random variables(page 47)
   2. Random sums of random variables(page 48)
3. Sampling distributions(page 48)
   1. Definitions(page 48)
   2. The sample mean(page 49)
   3. Central limit theorem(page 49)
   4. The law of large numbers(page 49)
4. Finite population(page 50)
5. Theorems(page 51)
   1. Theorems: the chi-square distribution(page 51)
   2. Theorems: the t distribution(page 51)
   3. Theorems: the F distribution(page 51)
6. Order statistics(page 52)
   1. Definition(page 52)
   2. The first order statistic(page 53)
   3. The n^th order statistic(page 53)
   4. The median(page 53)
   5. Joint distributions(page 53)
   6. Midrange and range(page 54)
   7. Uniform distribution: order statistics(page 54)
      1. Tolerance intervals(page 55)
   8. Normal distribution: order statistics(page 55)
      1. Expected value of normal order statistics(page 55)
7. Range and studentized range(page 57)
   1. Probability integral of the range(page 57)
   2. Percentage points, studentized range(page 57)

Discrete Probability Distributions(page 63)

1. Bernoulli distribution(page 63)
   1. Properties(page 63)
   2. Variates(page 64)
2. Binomial distribution(page 64)
   1. Properties(page 64)
   2. Variates(page 65)
   3. Tables(page 65)
3. Geometric distribution(page 70)
   1. Properties(page 71)
   2. Variates(page 71)
   3. Tables(page 71)
4. Hypergeometric distribution(page 72)
   1. Properties(page 73)
   2. Variates(page 73)
5. Multinomial distribution(page 74)
   1. Properties(page 74)
   2. Variates(page 74)
6. Negative binomial distribution(page 74)
   1. Properties(page 75)
   2. Variates(page 75)
   3. Tables(page 75)
7. Poisson distribution(page 76)
   1. Properties(page 76)
   2. Variates(page 76)
   3. Tables(page 76)
8. Rectangular (discrete uniform) distribution(page 79)
   1. Properties(page 79)

Continuous Probability Distributions(page 81)

1. Beta distribution(page 82)
   1. Properties(page 82)
   2. Probability density function(page 82)
   3. Related distributions(page 82)
2. Cauchy distribution(page 83)
   1. Properties(page 83)
   2. Probability density function(page 83)
   3. Related distributions(page 83)
3. Chi-square distribution(page 84)
   1. Properties(page 84)
   2. Probability density function(page 84)
   3. Related distributions(page 84)
   4. Critical values for chi-square distribution(page 85)
4. Erlang distribution(page 90)
   1. Properties(page 90)
   2. Probability density function(page 90)
   3. Related distributions(page 90)
5. Exponential distribution(page 90)
   1. Properties(page 90)
   2. Probability density function(page 90)
   3. Related distributions(page 91)
6. F distribution(page 92)
   1. Properties(page 92)
   2. Probability density function(page 92)
   3. Related distributions(page 92)
   4. Critical values for the F distribution(page 93)
7. Gamma distribution(page 99)
   1. Properties(page 99)
   2. Probability density function(page 99)
   3. Related distributions(page 99)
8. Lognormal distribution(page 100)
   1. Properties(page 100)
   2. Probability density function(page 100)
   3. Related distributions(page 100)
9. Normal distribution(page 101)
   1. Properties(page 101)
   2. Probability density function(page 101)
   3. Related distributions(page 101)
10. Normal distribution: multivariate(page 102)
    1. Properties(page 102)
    2. Probability density function(page 102)
11. Pareto distribution(page 103)
    1. Properties(page 103)
    2. Probability density function(page 103)
    3. Related distributions(page 103)
12. Rayleigh distribution(page 104)
    1. Properties(page 104)
    2. Probability density function(page 104)
    3. Related distributions(page 104)
13. t distribution(page 105)
    1. Properties(page 105)
    2. Probability density function(page 105)
    3. Related distributions(page 105)
    4. Critical values for the t distribution(page 105)
14. Triangular distribution(page 107)
    1. Properties(page 107)
    2. Probability density function(page 107)
15. Uniform distribution(page 108)
    1. Properties(page 108)
    2. Probability density function(page 108)
    3. Related distributions(page 108)
16. Weibull distribution(page 109)
    1. Properties(page 109)
    2. Probability density function(page 109)
    3. Related distributions(page 109)
17. Relationships among distributions(page 109)
    1. Other relationships among distributions(page 110)

Standard Normal Distribution(page 131)

1. Density function and related functions(page 131)
2. Critical values(page 141)
3. Tolerance factors for normal distributions(page 141)
   1. Tables of tolerance intervals(page 143)
4. Operating characteristic curves(page 144)
   1. One-sample Z test(page 144)
   2. Two-sample Z test(page 144)
5. Multivariate normal distribution(page 149)
6. Distribution of the correlation coefficient(page 149)
   1. Normal approximation(page 150)
   2. Zero coefficient for bivariate normal(page 150)

Estimation(page 153)

1. Definitions(page 153)
2. Cramer-Rao inequality(page 154)
3. Theorems(page 155)
4. The method of moments(page 156)
5. The likelihood function(page 156)
6. The method of maximum likelihood(page 156)
7. Invariance property of MLEs(page 157)
8. Different estimators(page 157)

Confidence Intervals(page 159)

1. Definitions(page 159)
2. Common critical values(page 159)
3. Sample size calculations(page 160)
4. Summary of common confidence intervals(page 161)
5. Other tests(page 162)
   1. Confidence interval for medians(page 162)
   2. Difference in medians(page 163)
6. Finite population correction factor(page 164)

Hypothesis Testing(page 165)

1. Introduction(page 165)
   1. Tables(page 166)
2. The Neyman-Pearson lemma(page 169)
3. Likelihood ratio tests(page 169)
4. Goodness of fit test(page 169)
5. Contingency tables(page 171)
6. Significance test in 2 x 2 contingency tables(page 172)
7. Critical values for testing outliers(page 173)

Regression Analysis(page 175)

1. Simple linear regression(page 175)
   1. Least squares estimates(page 176)
   2. Sum of squares(page 177)
   3. Inferences regarding regression coefficients(page 177)
   4. The mean response(page 178)
   5. Prediction interval(page 179)
   6. Analysis of variance table(page 179)
   7. Test for linearity of regression(page 179)
   8. Sample correlation coefficient(page 180)
   9. Example(page 180)
2. Multiple linear regression(page 182)
   1. Least squares estimates(page 182)

Nonparametric Statistics(page 183)

1. Friedman test for randomized block design(page 183)
2. Kendall's rank correlation coefficient(page 183)
   1. Kendall rank correlation coefficient table(page 184)
3. Kolmogorov-Smirnoff tests(page 185)
   1. One-sample Kolmogorov-Smirnoff test(page 185)
   2. Two-sample Kolmogorov-Smirnoff test(page 186)
   3. Tables for Kolmogorov-Smirnoff tests(page 188)
      1. Critical values, one-sample Kolmogorov-Smirnoff test(page 188)
      2. Critical values, two-sample Kolmogorov-Smirnoff test(page 189)
4. Kruskal-Wallis test(page 191)
   1. Tables for Kruskal-Wallis test(page 192)
5. The runs test(page 193)
   1. Tables for the runs test(page 193)
6. The sign test(page 203)
   1. Critical values for the sign test(page 203)
7. Spearman's rank correlation coefficient(page 204)
   1. Tables for Spearman's rank correlation(page 205)
8. Wilcoxon matched-pairs signed-ranks test(page 209)
9. Wilcoxon rank-sum (Mann-Whitney) test(page 210)
   1. Tables for Wilcoxon U statistic(page 211)
   2. Critical values for Wilcoxon U statistic(page 214)
10. Wilcoxon signed-rank test(page 218)

Miscellaneous topics(page 219)

1. Ceiling and floor functions(page 219)
2. Error functions(page 219)
   1. Special values(page 219)
3. Exponential function(page 220)
   1. Exponentiation(page 220)
   2. Definition of e^z(page 220)
   3. Derivative and integral of e^z(page 221)
   4. Circular functions and exponentials(page 221)
   5. Hyperbolic functions(page 221)
4. Factorials and Pochhammer's symbol(page 221)
5. Gamma function(page 222)
   1. Properties(page 224)
   2. Expansions(page 224)
   3. Special values(page 224)
6. Hypergeometric functions(page 225)
7. Logarithmic functions(page 225)
   1. Definition of the natural log(page 225)
   2. Special values(page 226)
   3. Logarithms to a base other than e(page 226)
   4. Relation of the logarithm to the exponential(page 226)
   5. Identities(page 226)
   6. Series expansions for the natural logarithm(page 226)
   7. Derivative and integration formulae(page 227)
8. Sums of powers of integers(page 227)
   1. Permutations(page 228)
   2. Combinations(page 229)

Notation(page 231)
Index(page 236)