Preface |
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xi | |
A note to the student (and instructor) |
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xvi | |
A note to the instructor (and student) |
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xviii | |
Acknowledgements |
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xxi | |
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1 | (6) |
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7 | (104) |
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9 | (34) |
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9 | (4) |
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Generalized binomial coefficients |
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13 | (2) |
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Combinatoric identities and the use of induction |
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15 | (3) |
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The binomial and multinomial theorems |
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18 | (10) |
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18 | (5) |
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An extension of the binomial theorem |
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23 | (4) |
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27 | (1) |
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The gamma and beta functions |
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28 | (8) |
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28 | (3) |
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31 | (5) |
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36 | (7) |
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Probability spaces and counting |
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43 | (30) |
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Introducing counting and occupancy problems |
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43 | (4) |
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47 | (11) |
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47 | (2) |
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49 | (9) |
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58 | (10) |
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58 | (1) |
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59 | (8) |
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67 | (1) |
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68 | (5) |
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Symmetric spaces and conditioning |
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73 | (38) |
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Applications with symmetric probability spaces |
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73 | (12) |
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Conditional probability and independence |
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85 | (12) |
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Total probability and Bayes' rule |
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87 | (6) |
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Extending the law of total probability |
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93 | (3) |
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Statistical paradoxes and fallacies |
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96 | (1) |
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The problem of the points |
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97 | (4) |
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97 | (2) |
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Further gambling problems |
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99 | (1) |
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Some historical references |
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100 | (1) |
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101 | (10) |
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Part II Discrete Random Variables |
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111 | (126) |
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Univariate random variables |
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113 | (52) |
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Definitions and properties |
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113 | (7) |
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Basic definitions and properties |
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113 | (4) |
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Further definitions and properties |
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117 | (3) |
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Discrete sampling schemes |
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120 | (20) |
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121 | (2) |
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123 | (2) |
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Geometric and negative binomial |
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125 | (3) |
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128 | (2) |
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130 | (3) |
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133 | (7) |
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140 | (1) |
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141 | (13) |
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141 | (2) |
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143 | (8) |
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151 | (3) |
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154 | (2) |
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156 | (9) |
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Multivariate random variables |
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165 | (32) |
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Multivariate density and distribution |
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165 | (6) |
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Joint cumulative distribution functions |
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166 | (2) |
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Joint probability mass and density functions |
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168 | (3) |
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Fundamental properties of multivariate random variables |
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171 | (11) |
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171 | (2) |
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173 | (1) |
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174 | (1) |
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175 | (1) |
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176 | (6) |
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Discrete sampling schemes |
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182 | (12) |
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182 | (6) |
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Multivariate hypergeometric |
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188 | (2) |
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Multivariate negative binomial |
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190 | (2) |
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Multivariate inverse hypergeometric |
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192 | (2) |
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194 | (3) |
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197 | (40) |
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197 | (2) |
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Use of exchangeable Bernoulli random variables |
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199 | (7) |
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202 | (4) |
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206 | (12) |
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Random variable decomposition |
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218 | (9) |
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Binomial, negative binomial and Poisson |
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218 | (2) |
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220 | (2) |
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222 | (5) |
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General linear combination of two random variables |
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227 | (5) |
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232 | (5) |
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Part III Continuous Random Variables |
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237 | (106) |
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Continuous univariate random variables |
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239 | (46) |
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Most prominent distributions |
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239 | (24) |
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Other popular distributions |
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263 | (6) |
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Univariate transformations |
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269 | (6) |
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Examples of one-to-one transformations |
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271 | (2) |
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Many-to-one transformations |
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273 | (2) |
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The probability integral transform |
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275 | (3) |
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276 | (1) |
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Kernel density estimation |
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277 | (1) |
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278 | (7) |
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Joint and conditional random variables |
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285 | (38) |
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285 | (5) |
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Conditional distributions |
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290 | (27) |
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291 | (1) |
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292 | (12) |
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304 | (6) |
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310 | (1) |
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311 | (1) |
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Computing probabilities via conditioning |
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312 | (5) |
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317 | (6) |
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Multivariate transformations |
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323 | (20) |
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323 | (6) |
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The t and F distributions |
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329 | (4) |
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Further aspects and important transformations |
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333 | (6) |
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339 | (4) |
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343 | (1) |
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343 | (92) |
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343 | (2) |
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Sets, functions and fundamental inequalities |
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345 | (5) |
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350 | (63) |
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351 | (1) |
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352 | (12) |
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364 | (18) |
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382 | (31) |
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413 | (22) |
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Neighborhoods and open sets |
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413 | (1) |
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Sequences, limits and continuity |
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414 | (2) |
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416 | (9) |
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425 | (10) |
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435 | (6) |
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441 | (10) |
References |
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451 | (10) |
Index |
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461 | |