Standard Probability and Statistics Tables

Zwillinger and Kokoska

Table of Contents

  1. Introduction(page 1)

    1. Background(page 1)
    2. Data sets(page 1)
    3. References(page 2)

  2. Summarizing Data(page 3)

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

  3. Probability(page 21)

    1. Algebra of sets(page 22)
    2. Combinatorial methods(page 24)
      1. The product rule for ordered pairs(page 24)
      2. The generalized product rule for k-tuples(page 25)
      3. Permutations(page 25)
      4. Circular permutations(page 25)
      5. Combinations (binomial coefficients)(page 25)
      6. Sample selection(page 26)
      7. Balls into cells(page 27)
      8. Multinomial coefficients(page 27)
      9. Arrangements and derangements(page 28)
    3. Probability(page 28)
      1. Relative frequency concept of probability(page 29)
      2. Axioms of probability (discrete sample space)(page 29)
      3. The probability of an event(page 29)
      4. Probability theorems(page 29)
      5. Probability and odds(page 30)
      6. Conditional probability(page 30)
      7. The multiplication rule(page 31)
      8. The law of total probability(page 31)
      9. Bayes' theorem(page 32)
      10. Independence(page 32)
    4. Random variables(page 33)
      1. Discrete random variables(page 33)
        1. Probability mass function(page 33)
        2. Cumulative distribution function(page 33)
      2. Continuous random variables(page 34)
        1. Probability density function(page 34)
        2. Cumulative distribution function(page 34)
      3. Random functions(page 34)
    5. Mathematical expectation(page 34)
      1. Expected value(page 34)
      2. Variance(page 35)
        1. Theorems(page 35)
      3. Moments(page 35)
        1. Moments about the origin(page 35)
        2. Moments about the mean(page 36)
        3. Factorial moments(page 36)
      4. Generating functions(page 36)
        1. Moment generating function(page 36)
        2. Factorial moment generating functions(page 37)
        3. Factorial moment generating function theorems(page 38)
        4. Cumulant generating function(page 38)
        5. Characteristic function(page 39)
    6. Multivariate distributions(page 39)
      1. Discrete case(page 39)
      2. Continuous case(page 39)
      3. Expectation(page 40)
      4. Moments(page 40)
      5. Marginal distributions(page 41)
      6. Independent random variables(page 42)
      7. Conditional distributions(page 42)
      8. Variance and covariance(page 43)
      9. Correlation coefficient(page 44)
      10. Moment generating function(page 44)
      11. Linear combination of random variables(page 45)
      12. Bivariate distribution(page 45)
        1. Joint probability distribution(page 45)
        2. Cumulative distribution function(page 46)
        3. Marginal distributions(page 47)
        4. Conditional distributions(page 47)
        5. Conditional expectation(page 48)
    7. Inequalities(page 48)

  4. Functions of Random Variables(page 51)

    1. Finding the probability distribution(page 52)
      1. Method of distribution functions(page 52)
      2. Method of transformations (one variable)(page 53)
      3. Method of transformations (many variables)(page 54)
      4. Method of moment generating functions(page 55)
    2. Sums of random variables(page 56)
      1. Deterministic sums of random variables(page 56)
      2. Random sums of random variables(page 57)
    3. Sampling distributions(page 57)
      1. Definitions(page 57)
      2. The sample mean(page 58)
      3. Central limit theorem(page 58)
      4. The law of large numbers(page 58)
      5. Laws of the iterated logarithm(page 59)
    4. Finite population(page 59)
    5. Theorems(page 60)
      1. Theorems: the chi-square distribution(page 60)
      2. Theorems: the t distribution(page 60)
      3. Theorems: the F distribution(page 61)
    6. Order statistics(page 61)
      1. Definition(page 61)
      2. The first order statistic(page 62)
      3. The nth order statistic(page 62)
      4. The median(page 62)
      5. Joint distributions(page 62)
      6. Midrange and range(page 63)
      7. Uniform distribution: order statistics(page 63)
        1. Tolerance intervals(page 64)
      8. Normal distribution: order statistics(page 64)
        1. Expected value of normal order statistics(page 64)
        2. Variances and covariances of order statistics(page 66)
    7. Range and studentized range(page 68)
      1. Probability integral of the range(page 68)
      2. Percentage points, studentized range(page 76)

  5. Discrete Probability Distributions(page 81)

    1. Bernoulli distribution(page 82)
      1. Properties(page 82)
      2. Variates(page 83)
    2. Beta binomial distribution(page 83)
      1. Properties(page 83)
      2. Variates(page 84)
    3. Beta Pascal distribution(page 84)
      1. Properties(page 84)
    4. Binomial distribution(page 84)
      1. Properties(page 84)
      2. Variates(page 85)
      3. Tables(page 85)
    5. Geometric distribution(page 92)
      1. Properties(page 92)
      2. Variates(page 92)
      3. Tables(page 92)
    6. Hypergeometric distribution(page 94)
      1. Properties(page 94)
      2. Variates(page 94)
      3. Tables(page 95)
    7. Multinomial distribution(page 101)
      1. Properties(page 101)
      2. Variates(page 101)
    8. Negative binomial distribution(page 101)
      1. Properties(page 101)
        1. Alternative characterization(page 102)
      2. Variates(page 102)
      3. Tables(page 103)
    9. Poisson distribution(page 103)
      1. Properties(page 104)
      2. Variates(page 104)
      3. Tables(page 104)
    10. Rectangular (discrete uniform) distribution(page 110)
      1. Properties(page 110)

  6. Continuous Probability Distributions(page 111)

    1. Arcsin distribution(page 114)
      1. Properties(page 114)
      2. Probability density function(page 114)
    2. Beta distribution(page 115)
      1. Properties(page 115)
      2. Probability density function(page 116)
      3. Related distributions(page 116)
    3. Cauchy distribution(page 117)
      1. Properties(page 117)
      2. Probability density function(page 117)
      3. Related distributions(page 117)
    4. Chi-square distribution(page 118)
      1. Properties(page 118)
      2. Probability density function(page 119)
      3. Related distributions(page 119)
      4. Critical values for chi-square distribution(page 120)
      5. Percentage points, chi-square over dof(page 125)
    5. Erlang distribution(page 125)
      1. Properties(page 125)
      2. Probability density function(page 126)
      3. Related distributions(page 126)
    6. Exponential distribution(page 126)
      1. Properties(page 126)
      2. Probability density function(page 126)
      3. Related distributions(page 127)
    7. Extreme-value distribution(page 128)
      1. Properties(page 128)
      2. Probability density function(page 128)
      3. Related distributions(page 128)
    8. F distribution(page 129)
      1. Properties(page 129)
      2. Probability density function(page 130)
      3. Related distributions(page 130)
      4. Critical values for the F distribution(page 131)
    9. Gamma distribution(page 138)
      1. Properties(page 138)
      2. Probability density function(page 138)
      3. Related distributions(page 138)
    10. Half--normal distribution(page 139)
      1. Properties(page 139)
      2. Probability density function(page 139)
    11. Inverse Gaussian (Wald) distribution(page 140)
      1. Properties(page 140)
      2. Probability density function(page 140)
      3. Related distributions(page 140)
    12. Laplace distribution(page 141)
      1. Properties(page 141)
      2. Probability density function(page 141)
      3. Related distributions(page 141)
    13. Logistic distribution(page 142)
      1. Properties(page 142)
      2. Probability density function(page 142)
      3. Related distributions(page 143)
    14. Lognormal distribution(page 143)
      1. Properties(page 143)
      2. Probability density function(page 144)
      3. Related distributions(page 144)
    15. Noncentral chi-square distribution(page 145)
      1. Properties(page 145)
      2. Probability density function(page 145)
      3. Related distributions(page 146)
    16. Noncentral F distribution(page 146)
      1. Properties(page 146)
      2. Probability density function(page 146)
      3. Related distributions(page 146)
    17. Noncentral t distribution(page 147)
      1. Properties(page 147)
      2. Probability density function(page 147)
      3. Related distributions(page 148)
    18. Normal distribution(page 148)
      1. Properties(page 148)
      2. Probability density function(page 148)
      3. Related distributions(page 149)
    19. Normal distribution: multivariate(page 150)
      1. Properties(page 150)
      2. Probability density function(page 150)
    20. Pareto distribution(page 151)
      1. Properties(page 151)
      2. Probability density function(page 151)
      3. Related distributions(page 151)
    21. Power function distribution(page 152)
      1. Properties(page 152)
      2. Probability density function(page 152)
      3. Related distributions(page 153)
    22. Rayleigh distribution(page 154)
      1. Properties(page 154)
      2. Probability density function(page 154)
      3. Related distributions(page 155)
    23. t distribution(page 155)
      1. Properties(page 155)
      2. Probability density function(page 155)
      3. Related distributions(page 155)
      4. Critical values for the t distribution(page 156)
    24. Triangular distribution(page 158)
      1. Properties(page 158)
      2. Probability density function(page 158)
    25. Uniform distribution(page 159)
      1. Properties(page 159)
      2. Probability density function(page 159)
      3. Related distributions(page 159)
    26. Weibull distribution(page 160)
      1. Properties(page 160)
      2. Probability density function(page 160)
      3. Related distributions(page 160)
    27. Relationships among distributions(page 161)
      1. Other relationships among distributions(page 161)

  7. Standard Normal Distribution(page 165)

    1. Density function and related functions(page 165)
    2. Critical values(page 175)
    3. Tolerance factors for normal distributions(page 175)
      1. Tables of tolerance intervals(page 177)
    4. Operating characteristic curves(page 178)
      1. One-sample Z test(page 178)
      2. Two-sample Z test(page 178)
    5. Multivariate normal distribution(page 181)
    6. Distribution of the correlation coefficient(page 182)
      1. Normal approximation(page 183)
      2. Zero coefficient for bivariate normal(page 183)
    7. Circular normal probabilities(page 185)
    8. Circular error probabilities(page 186)

  8. Estimation(page 187)

    1. Definitions(page 187)
    2. Cramer--Rao inequality(page 188)
    3. Theorems(page 189)
    4. The method of moments(page 190)
    5. The likelihood function(page 190)
    6. The method of maximum likelihood(page 191)
    7. Invariance property of MLEs(page 191)
    8. Different estimators(page 191)
    9. Estimators for small samples(page 193)
    10. Estimators for large samples(page 194)

  9. Confidence Intervals(page 195)

    1. Definitions(page 195)
    2. Common critical values(page 196)
    3. Sample size calculations(page 196)
    4. Summary of common confidence intervals(page 198)
    5. Confidence intervals: one sample(page 198)
      1. Mean of normal population, known variance(page 198)
      2. Mean of normal population, unknown var(page 199)
      3. Variance of normal population(page 200)
      4. Success in binomial experiments(page 201)
      5. Confidence interval for percentiles(page 201)
      6. Confidence interval for medians(page 201)
        1. Table of confidence interval for medians(page 201)
      7. Confidence interval for Poisson distribution(page 202)
      8. Confidence interval for binomial distribution(page 205)
    6. Confidence intervals: two samples(page 222)
      1. Difference in means, known variances(page 222)
      2. Difference in means, equal unknown var(page 222)
      3. Difference in means, unequal unknown var(page 222)
      4. Difference in means, paired observations(page 223)
      5. Ratio of variances(page 223)
      6. Difference in success probabilities(page 223)
      7. Difference in medians(page 224)
    7. Finite population correction factor(page 225)

  10. Hypothesis Testing(page 227)

    1. Introduction(page 227)
      1. Tables(page 228)
    2. The Neyman--Pearson lemma(page 231)
    3. Likelihood ratio tests(page 232)
    4. Goodness of fit test(page 232)
    5. Contingency tables(page 233)
    6. Bartlett's test(page 234)
      1. Approximate test procedure(page 235)
      2. Tables for Bartlett's test(page 235)
    7. Cochran's test(page 238)
      1. Tables for Cochran's test(page 238)
    8. Number of observations required(page 241)
    9. Critical values for testing outliers(page 242)
    10. Significance test in 2 x 2 contingency tables(page 244)
    11. Determining values in Bernoulli trials(page 260)

  11. Regression Analysis(page 261)

    1. Simple linear regression(page 261)
      1. Least squares estimates(page 263)
      2. Sum of squares(page 263)
      3. Inferences regarding regression coefficients(page 264)
      4. The mean response(page 264)
      5. Prediction interval(page 265)
      6. Analysis of variance table(page 266)
      7. Test for linearity of regression(page 266)
      8. Sample correlation coefficient(page 266)
      9. Example(page 267)
    2. Multiple linear regression(page 268)
      1. Least squares estimates(page 269)
      2. Sum of squares(page 269)
      3. Inferences regarding regression coefficients(page 270)
      4. The mean response(page 270)
      5. Prediction interval(page 271)
      6. Analysis of variance table(page 271)
      7. Sequential sum of squares(page 272)
      8. Partial F test(page 272)
      9. Residual analysis(page 273)
      10. Example(page 274)
    3. Orthogonal polynomials(page 275)
      1. Tables for orthogonal polynomials(page 278)

  12. Analysis of Variance(page 281)

    1. One-way anova(page 282)
      1. Sum of squares(page 282)
      2. Properties(page 283)
      3. Analysis of variance table(page 283)
      4. Multiple comparison procedures(page 284)
        1. Tukey's procedure(page 284)
        2. Duncan's multiple range test(page 284)
        3. Duncan's multiple range test(page 285)
        4. Dunnett's procedure(page 289)
        5. Tables for Dunnett's procedure(page 289)
      5. Contrasts(page 291)
      6. Example(page 292)
    2. Two-way anova(page 294)
      1. One observation per cell(page 294)
        1. Models and assumptions(page 294)
        2. Sum of squares(page 294)
        3. Mean squares and properties(page 295)
      2. Analysis of variance table(page 295)
      3. Nested classifications with equal samples(page 296)
        1. Models and assumptions(page 296)
        2. Sum of squares(page 297)
        3. Mean squares and properties(page 297)
        4. Analysis of variance table(page 298)
      4. Nested classifications with unequal samples(page 299)
        1. Models and assumptions(page 299)
        2. Sum of squares(page 300)
        3. Mean squares and properties(page 301)
        4. Analysis of variance table(page 302)
      5. Two-factor experiments(page 303)
        1. Models and assumptions(page 303)
        2. Sum of squares(page 304)
        3. Mean squares and properties(page 304)
        4. Analysis of variance table(page 305)
      6. Example(page 306)
    3. Three-factor experiments(page 308)
      1. Models and assumptions(page 308)
      2. Sum of squares(page 309)
      3. Mean squares and properties(page 311)
      4. Analysis of variance table(page 314)
    4. Manova(page 317)
    5. Factor analysis(page 317)
    6. Latin square design(page 319)
      1. Models and assumptions(page 319)
      2. Sum of squares(page 320)
      3. Mean squares and properties(page 320)
      4. Analysis of variance table(page 321)

  13. Experimental Design(page 323)

    1. Latin squares(page 323)
    2. Graeco--Latin squares(page 324)
    3. Block designs(page 325)
    4. Factorial experimentation: 2 factors(page 327)
    5. 2r Factorial experiments(page 328)
    6. Confounding in 2n factorial experiments(page 330)
    7. Tables for design of experiments(page 330)
      1. Plans of factorial experiments confounded(page 331)
      2. Plans of 2n factorials in fractional replication(page 334)
      3. Plans of incomplete block designs(page 336)
      4. Interactions in factorial designs(page 338)
    8. References(page 341)

  14. Nonparametric Statistics(page 343)

    1. Friedman test for randomized block design(page 344)
    2. Kendall's rank correlation coefficient(page 344)
      1. Kendall rank correlation coefficient table(page 345)
    3. Kolmogorov--Smirnoff tests(page 346)
      1. One-sample Kolmogorov--Smirnoff test(page 346)
      2. Two-sample Kolmogorov--Smirnoff test(page 346)
      3. Tables for Kolmogorov--Smirnoff tests(page 348)
        1. Critical values, one-sample Kolmogorov--Smirnoff test(page 348)
        2. Critical values, two-sample Kolmogorov--Smirnoff test(page 349)
    4. Kruskal--Wallis test(page 351)
      1. Tables for Kruskal--Wallis test(page 352)
    5. The runs test(page 353)
      1. Tables for the runs test(page 354)
    6. The sign test(page 365)
      1. Critical values for the sign test(page 365)
    7. Spearman's rank correlation coefficient(page 366)
      1. Tables for Spearman's rank correlation(page 367)
    8. Wilcoxon matched-pairs signed-ranks test(page 371)
    9. Wilcoxon rank--sum (Mann--Whitney) test(page 372)
      1. Tables for Wilcoxon U statistic(page 373)
      2. Critical values for Wilcoxon U statistic(page 377)
    10. Wilcoxon signed-rank test(page 381)

  15. Quality Control and Risk Analysis(page 383)

    1. Quality assurance(page 383)
      1. Control charts(page 383)
      2. Abnormal distributions of points(page 386)
    2. Acceptance sampling(page 386)
      1. Sequential sampling(page 387)
        1. Sequential probability ratio tests(page 389)
        2. Two-sided mean test(page 390)
        3. One-sided variance test(page 390)
    3. Reliability(page 391)
      1. Failure time distributions(page 391)
        1. Use of the exponential distribution(page 392)
    4. Risk analysis and decision rules(page 392)

  16. General Linear Models(page 397)

    1. Notation(page 398)
    2. The general linear model(page 398)
      1. The simple linear regression model(page 398)
      2. Multiple linear regression(page 400)
      3. One-way analysis of variance(page 401)
      4. Two-way analysis of variance(page 402)
      5. Analysis of covariance(page 403)
    3. Summary of rules for matrix operations(page 404)
      1. Linear combinations(page 404)
      2. Determinants(page 404)
      3. Inverse of a partitioned matrix(page 405)
        1. Symmetric case(page 405)
      4. Eigenvalues(page 405)
      5. Differentiation involving vectors/matrices(page 406)
        1. Definitions(page 406)
        2. Properties(page 406)
      6. Additional definitions and properties(page 407)
    4. Quadratic forms ... (page 408)
      1. Multivariate distributions(page 408)
      2. The principle of least squares(page 409)
      3. Minimum variance unbiased estimates(page 409)
    5. General linear hypothesis of full rank(page 410)
      1. Notation(page 410)
      2. Simple linear regression(page 410)
      3. Analysis of variance, one-way anova(page 411)
      4. Multiple linear regression(page 412)
      5. Randomized blocks (one observation per cell)(page 413)
      6. Quadratic form due to hypothesis(page 414)
      7. Sum of squares due to error(page 415)
      8. Summary(page 416)
      9. Computation procedure for hypothesis testing(page 416)
      10. Regression significance test(page 417)
      11. Alternate form of the distribution(page 417)
    6. General linear model of less than full rank(page 417)
      1. Estimable function and estimability(page 418)
      2. Linear hypothesis model of less than full rank(page 420)
        1. Sum of squares due to error(page 420)
        2. Sum of squares due to hypothesis(page 421)
      3. Constraints and conditions(page 422)

  17. Miscellaneous Topics(page 423)

    1. Geometric probability(page 425)
    2. Information and communication theory(page 427)
      1. Discrete entropy(page 427)
      2. Continuous entropy(page 429)
      3. Channel capacity(page 429)
      4. Shannon's theorem(page 430)
    3. Kalman filtering(page 431)
      1. Extended Kalman filtering(page 432)
    4. Large deviations (theory of rare events)(page 432)
      1. Theory(page 432)
      2. Sample rate functions(page 433)
      3. Example: Insurance company(page 433)
    5. Markov chains(page 434)
      1. Transition function(page 434)
      2. Transition matrix(page 435)
      3. Recurrence(page 435)
      4. Stationary distributions(page 436)
      5. Random walks(page 437)
      6. Ehrenfest chain(page 437)
    6. Martingales(page 438)
      1. Examples of martingales(page 438)
    7. Measure theoretical probability(page 438)
    8. Monte Carlo integration techniques(page 439)
      1. Importance sampling(page 440)
      2. Hit-or-miss Monte Carlo method(page 440)
    9. Queuing theory(page 441)
      1. M/M/1 queue(page 442)
      2. M/M/1/K queue(page 443)
      3. M/M/2 queue(page 443)
      4. M/M/c queue(page 444)
      5. M/M/c/c queue(page 444)
      6. M/M/c/K queue(page 444)
      7. M/M/infty queue(page 445)
      8. M/E_k/1 queue(page 445)
      9. M/D/1 queue(page 445)
    10. Random matrix eigenvalues(page 446)
      1. Random matrix products(page 449)
        1. Vibonacci numbers(page 449)
    11. Random number generation(page 450)
      1. Pseudorandom number generation(page 450)
        1. Linear congruential generators(page 450)
        2. Shift register generators(page 451)
        3. Lagged-Fibonacci generators(page 451)
      2. Generating nonuniform random variables(page 452)
        1. Discrete random variables(page 454)
        2. Testing pseudorandom numbers(page 455)
      3. References(page 455)
    12. Resampling methods(page 455)
    13. Self-similar processes(page 456)
      1. Definitions(page 456)
      2. Self-similar processes(page 457)
    14. Signal processing(page 457)
      1. Estimation(page 457)
      2. Matched filtering (Wiener filter)(page 458)
      3. Median filter(page 458)
      4. Mean filter(page 458)
      5. Spectral decompositions(page 459)
    15. Stochastic calculus(page 459)
      1. Brownian motion (Wiener processes)(page 459)
      2. Brownian motion expectations(page 460)
      3. Ito lemma(page 462)
      4. Stochastic integration(page 462)
      5. Stochastic differential equations(page 463)
      6. Motion in a domain(page 464)
      7. Option Pricing(page 465)
    16. Classic and interesting problems(page 466)
      1. Approximating a distribution(page 466)
      2. Averages over vectors(page 466)
      3. Bertrand's box ``paradox''(page 466)
      4. Bertrand's circle ``paradox''(page 467)
      5. Bingo cards: nontransitive(page 467)
      6. Birthday problem(page 468)
      7. Buffon's needle problem(page 468)
      8. Card problems(page 468)
        1. Shuffling cards(page 468)
        2. Card games(page 469)
      9. Coin problems(page 470)
        1. Even odds from a biased coin(page 470)
        2. Two heads in a row(page 471)
      10. Coupon collectors problem(page 471)
      11. Dice problems(page 472)
        1. Dice: nontransitive (Efron)(page 472)
        2. Dice: distribution of sums(page 472)
        3. Dice: same distribution(page 472)
      12. Ehrenfest urn model(page 473)
      13. Envelope problem ``paradox''(page 473)
      14. Gender distributions(page 473)
      15. Holtzmark distribution: stars in the galaxy(page 474)
      16. Large-scale testing(page 474)
        1. Infrequent success(page 474)
        2. Pooling of blood samples(page 475)
      17. Leading digit distribution(page 475)
        1. Ratio of uniform numbers(page 475)
        2. Benford's law(page 476)
      18. Lotteries(page 476)
      19. Match box problem(page 477)
      20. Maximum entropy distributions(page 477)
      21. Monte Hall problem(page 477)
      22. Multi-armed bandit problem(page 478)
      23. Parking problem(page 478)
      24. Passage problems(page 478)
      25. Proofreading mistakes(page 479)
      26. Raisin cookie problem(page 479)
      27. Random sequences(page 480)
        1. Long runs(page 480)
        2. Waiting times: two types of characters(page 480)
        3. Waiting times: many types of characters(page 480)
        4. First random sequence(page 481)
      28. Random walks(page 482)
        1. Random walk on a grid(page 482)
        2. Random walk in two dimensions(page 483)
        3. Random walk in three dimensions(page 483)
        4. Self-avoiding walks(page 484)
        5. Gambler's ruin problem(page 484)
      29. Relatively prime integers(page 485)
      30. Roots of a random polynomial(page 485)
      31. Roots of a random quadratic(page 486)
      32. Simpson paradox(page 486)
      33. Secretary call problem(page 487)
      34. Waiting for a bus(page 487)
    17. Electronic resources(page 487)
      1. Statlib(page 487)
      2. Uniform resource locators(page 488)
      3. Interactive demonstrations and tutorials(page 489)
      4. Textbooks, manuals, and journals(page 491)
      5. Free statistical software packages(page 493)
      6. Demonstration statistical software packages(page 496)
    18. Tables(page 497)
      1. Random deviates(page 497)
      2. Permutations(page 500)
      3. Combinations(page 500)

  18. Special Functions(page 505)

    1. Bessel functions(page 506)
      1. Differential equation(page 506)
      2. Series expansions(page 507)
      3. Recurrence relations(page 507)
      4. Behavior as z ==> 0(page 508)
      5. Integral representations(page 508)
      6. Fourier expansion(page 509)
      7. Asymptotic expansions(page 509)
      8. Half-order Bessel functions(page 509)
      9. Modified Bessel functions(page 509)
        1. Relation to ordinary Bessel functions(page 510)
        2. Recursion relations(page 511)
        3. Integrals(page 511)
    2. Beta function(page 511)
      1. Other integrals(page 511)
      2. Properties(page 511)
    3. Ceiling and floor functions(page 512)
    4. Delta function(page 512)
    5. Error functions(page 512)
      1. Expansions(page 513)
      2. Special values(page 513)
    6. Exponential function(page 513)
      1. Exponentiation(page 513)
      2. Definition of ez(page 514)
      3. Derivative and integral of ez(page 514)
      4. Circular functions and exponentials(page 514)
      5. Hyperbolic functions(page 515)
    7. Factorials and Pochhammer's symbol(page 515)
    8. Gamma function(page 515)
      1. Other integrals for the gamma function(page 516)
      2. Properties(page 517)
      3. Expansions(page 518)
      4. Special values(page 518)
      5. Digamma function(page 518)
      6. Incomplete gamma functions(page 519)
    9. Hypergeometric functions(page 520)
      1. Generalized hypergeometric function(page 520)
      2. Gauss hypergeometric function(page 520)
        1. Special cases(page 520)
        2. Functional relations(page 521)
      3. Confluent hypergeometric functions(page 521)
    10. Logarithmic functions(page 521)
      1. Definition of the natural log(page 521)
      2. Special values(page 521)
      3. Logarithms to a base other than e(page 522)
      4. Relation of the logarithm to the exponential(page 522)
      5. Identities(page 522)
      6. Series expansions for the natural logarithm(page 522)
      7. Derivative and integration formulae(page 522)
    11. Partitions(page 523)
    12. Signum function(page 523)
    13. Stirling numbers(page 523)
      1. Stirling numbers(page 523)
      2. Stirling cycle numbers(page 525)
    14. Sums of powers of integers(page 525)
    15. Tables of orthogonal polynomials(page 527)
    16. References(page 528)

Notation(page 528)
Index(page 536)