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Multiplication Rule -- Example 5
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Probability
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- 1 Introduction
- 2 Monty Hall (Let's Make a Deal) Problem
- 3 The birthday problem
- 4 Univariate data set
- 5 Bivariate data set
- 6 Bivariate data set
- 7 Multivariate data set
- 8 Simple linear regression
- 9 Enumeration vs. counting
- 10 Multiplication rule
- 11 Multiplication Rule -- Example 1
- 12 Multiplication Rule -- Example 2
- 13 Multiplication Rule -- Example 3
- 14 Multiplication Rule -- Example 4
- 15 Multiplication Rule -- Example 5
- 16 Multiplication Rule -- Example 6
- 17 Permutations
- 18 Permutations -- Example 1
- 19 Permutations -- Example 2
- 20 Circular Permutations -- Example 1
- 21 Circular Permutations -- Example 2
- 22 Nondistinct permutations
- 23 Nondistinct Permutations -- Example 1
- 24 Nondistinct Permutations -- Example 2
- 25 Nondistinct Permutations -- Example 3
- 26 Nondistinct Permutations -- Example 4
- 27 Combinations
- 28 Combinations -- Example 1
- 29 Combinations -- Example 2
- 30 Combinations -- Example 3
- 31 Combinations -- notes
- 32 Partitioning -- Example 1
- 33 Partitioning -- Example 2
- 34 Partitioning -- Example 3
- 35 Partitioning -- Example 4
- 36 Counting techniques -- unifying example
- 37 Set theory
- 38 Operations on sets
- 39 Venn diagrams -- Example 1
- 40 Venn diagrams -- Example 2
- 41 Set theory -- notes
- 42 Set theory -- application
- 43 Probability introduction
- 44 Random experiments
- 45 Sample spaces
- 46 Sample space classification
- 47 Sample space subsets -- events
- 48 Approaches for calculating probabilities
- 49 Relative frequency approach to estimating probability
- 50 Relative frequency approach (limiting case)
- 51 Subjective approach to estimating probability
- 52 Classical approach to calculating probability
- 53 Probability axioms
- 54 Complementary probability
- 55 Probability result concerning subsets
- 56 Probability of unions of events
- 57 Computing probabilities -- Example 1
- 58 Computing probabilities -- Example 2
- 59 Computing probabilities -- Example 3
- 60 Computing probabilities -- Example 4
- 61 Computing probabilities -- Example 5
- 62 Computing probabilities -- Example 6
- 63 Computing probabilities -- Example 7
- 64 Computing probabilities -- Example 8
- 65 Computing probabilities -- Example 9
- 66 Computing probabilities -- Example 10
- 67 Computing probabilities -- Example 11
- 68 Conditional probability -- Example 1
- 69 Conditional probability -- Example 2
- 70 Conditional probability notes
- 71 Conditional probability -- Example 3
- 72 Rule of elimination -- law of total probability
- 73 Conditional probability -- Example 4
- 74 Rule of Bayes
- 75 Rule of Bayes -- Example 1
- 76 Rule of Bayes -- Example 2
- 77 Independence
- 78 Independence -- Example 0
- 79 Independence -- Example 1
- 80 Mutual independence
- 81 Independence in a series system
- 82 Independence in a parallel system
- 83 Independence -- Example 2
- 84 Random variables
- 85 Discrete random variable definition
- 86 Discrete random variables -- Example 1
- 87 Probability mass functions
- 88 Discrete random variables -- Example 2
- 89 Discrete random variables -- Example 3
- 90 Discrete random variables -- Example 4
- 91 Discrete random variables -- Example 5
- 92 Discrete random variables -- Example 6
- 93 Discrete random variables -- Example 7
- 94 Discrete random variables -- Example 8
- 95 Discrete random variable summary
- 96 Continuous random variables introduction
- 97 Probability density functions
- 98 Continuous random variables -- Example 0
- 99 Continuous random variables -- Example 1
- 100 Continuous random variables -- Example 2
- 101 Classifying random variables
- 102 Mixed random variables
- 103 Continuous random variables summary
- 104 Cumulative distribution function definition
- 105 Cumulative distribution function notes
- 106 Cumulative distribution function conversion
- 107 Cumulative distribution functions -- Example 1
- 108 Cumulative distribution functions -- Example 2
- 109 Cumulative distribution functions -- Example 3
- 110 Cumulative distribution functions -- Example 4
- 111 Cumulative distribution functions -- Example 5
- 112 Cumulative distribution function of a mixed random variable
- 113 Cumulative distribution function topics
- 114 Percentiles
- 115 Percentiles -- Example 1
- 116 Percentiles -- Example 2
- 117 Percentiles -- Example 3
- 118 Random variate generation
- 119 Random variate generation -- Example 1
- 120 Random variate generation -- Example 2
- 121 Transformations of random variables
- 122 Transformations of random variables -- Example 1
- 123 Transformations of random variables -- Example 2
- 124 APPL introduction
- 125 APPL -- Example 1
- 126 APPL data structure
- 127 APPL -- Example 2
- 128 APPL -- Example 3
- 129 APPL -- Example 4
- 130 Mixtures
- 131 Mixtures -- Example 1
- 132 Mixtures application
- 133 Continuous mixtures
- 134 Expectation
- 135 Expectation -- Example 1
- 136 Expectation -- Example 2
- 137 Expectation -- Example 3
- 138 Expectation -- Example 4
- 139 Expectation -- Example 5
- 140 Expectation -- Example 6
- 141 Expectation -- Example 7
- 142 Expectation -- Example 8
- 143 Expectation -- Example 9
- 144 Measures of central tendency
- 145 Measures of central tendency -- Example 1
- 146 Population mode definition
- 147 Population mean summary
- 148 Expectation topics
- 149 Expectation of a constant
- 150 Expectation of a constant times a random variable
- 151 Expectation of a function of a random variable -- Example 1
- 152 Expectation of a function of a random variable -- Example 1
- 153 Expectation of the function of a random variable -- Example 1
- 154 Expectation of a constant times a function of a random variable
- 155 Expectation of the sum of two functions of a random variable
- 156 Population variance definition
- 157 Notes on population variance
- 158 Population variance shortcut formula
- 159 Population variance -- Example 1
- 160 Population variance of aX+b
- 161 Population variance corollaries
- 162 Moment definition
- 163 Standardized random variables
- 164 Skewness
- 165 Kurtosis
- 166 Skewness and kurtosis -- Example 1
- 167 Moment generating function definition
- 168 Using moment generating functions to generate moments
- 169 Moment generating functions -- Example 1
- 170 Moment generating functions -- Example 2
- 171 Characteristic functions
- 172 Conditional expectation
- 173 Markov's inequality
- 174 Markov's inequality -- Example 1
- 175 Chebyshev's inequality
- 176 Chebyshev's inequality -- Example 1
- 177 Common discrete distributions
- 178 Bernoulli distribution definition
- 179 Bernoulli trials
- 180 Bernoulli distribution moments
- 181 Bernoulli distribution summary
- 182 Binomial distribution definition
- 183 Binomial distribution notes
- 184 Binomial distribution mean
- 185 Binomial distribution moments
- 186 Binomial distribution shape
- 187 Binomial distribution -- Example 1
- 188 Binomial distribution -- Example 2
- 189 Binomial distribution calculations in R
- 190 Binomial distribution -- Example 3
- 191 Binomial distribution -- Example 4
- 192 Binomial distribution -- Example 5
- 193 Binomial distribution summary
- 194 Geometric distribution definition
- 195 Geometric distribution existence conditions
- 196 Geometric distribution cumulative distribution function
- 197 Geometric distribution memoryless property
- 198 Geometric distribution moment generating function
- 199 Geometric distribution population mean
- 200 Geometric distribution moments
- 201 Geometric distribution -- Example 1
- 202 Geometric distribution definition
- 203 Geometric distribution -- Example 2
- 204 Negative binomial distribution
- 205 Negative binomial moment generating function
- 206 Negative binomial distribution -- Example 1
- 207 Negative binomial distribution
- 208 Negative binomial distribution -- Example 2
- 209 Poisson distribution introduction
- 210 Poisson approximation to the binomial distribution
- 211 Poisson distribution definition
- 212 Poisson distribution moment generating function
- 213 Poisson distribution -- Example 1
- 214 Poisson processes introduction
- 215 Poisson processes illustrations
- 216 Poisson process notation
- 217 Poisson process counting function
- 218 Poisson processes -- Example 1
- 219 Poisson process time between arrivals
- 220 Poisson process superpositioning
- 221 Poisson process decomposition
- 222 Poisson processes and order statistics
- 223 Poisson process summary
- 224 Poisson distribution -- Horse kick data
- 225 Hypergeometric distribution introduction
- 226 Hypergeometric distribution
- 227 Hypergeometric distribution support
- 228 Hypergeometric distribution moments
- 229 Hypergeometric distribution -- Example 1
- 230 Discrete uniform distribution
- 231 Discrete uniform distribution -- Example 1
- 232 Benford's law -- Benford distribution
- 233 Zipf distribution
- 234 Zipf distribution -- Example 1
- 235 Mixture distribution
- 236 Discrete distribution summary
- 237 Common continuous distributions
- 238 Uniform distribution
- 239 Uniform distribution cumulative distribution function
- 240 Uniform distribution moment generating function
- 241 Uniform distribution moments
- 242 Uniform distribution -- Example 1
- 243 Uniform distribution -- Example 2
- 244 Uniform distribution -- Example 3
- 245 Uniform distribution -- Example 4
- 246 Uniform distribution -- Example 5
- 247 Uniform distribution -- Example 6
- 248 Exponential distribution definition
- 249 Exponential distribution rate parameter
- 250 Exponential distribution cumulative distribution function
- 251 Exponential distribution memoryless property
- 252 Exponential distribution moment generating function
- 253 Exponential distribution moments
- 254 Gamma function
- 255 Exponential distribution -- Example 1
- 256 Exponential distribution -- Example 2
- 257 Exponential distribution -- Example 3
- 258 Exponential distribution -- Example 4
- 259 Exponential distribution summary
- 260 Gamma Distribution Definition
- 261 Gamma distribution moment generating function
- 262 Gamma distribution moments
- 263 Gamma distribution special cases
- 264 Gamma distribution -- Example 1
- 265 Gamma distribution summary
- 266 Normal distribution introduction
- 267 Normal distribution history
- 268 Normal distribution properties
- 269 Normal distribution computations
- 270 Normal distribution moment generating function
- 271 Normal distribution Y = a + bX result
- 272 Normal distribution Z = (X - mu) / sigma
- 273 Normal distribution Y = ((X - mu) / sigma) ^ 2 result
- 274 Normal distribution -- Example 1
- 275 Normal distribution -- Example 2
- 276 Other continuous distributions
- 277 Beta distribution
- 278 Beta function
- 279 Beta distribution mean
- 280 Beta distribution moments
- 281 Beta distribution -- Example 1
- 282 Beta distribution -- Example 2
- 283 Triangular distribution
- 284 Triangular distribution cumulative distribution function
- 285 Triangular distribution moments
- 286 Triangular distribution -- Example 1
- 287 Weibull Distribution
- 288 Weibull distribution moments
- 289 Weibull distribution -- Example 1
- 290 Continuous distributions summary
- 291 Multivariate distributions introduction
- 292 Bivariate distribution introduction
- 293 Bivariate distributions -- Automobile illustration
- 294 Bivariate distribution definition
- 295 Bivariate distribution pmf/pdf
- 296 Bivariate distributions -- Example 1
- 297 Bivariate distributions -- Example 2
- 298 Bivariate distributions -- Example 3
- 299 Bivariate distributions -- Example 4
- 300 Bivariate distributions -- Example 5
- 301 Bivariate distributions notation
- 302 Bivariate distributions cumulative distribution functions
- 303 Bivariate distributions cumulative distribution functions -- Example 1
- 304 Bivariate distributions; marginal distributions
- 305 Bivariate distributions; marginal distributions -- Example 1
- 306 Bivariate distributions; marginal distributions -- Example 2
- 307 Bivariate distributions; marginal distributions -- Example 3
- 308 Bivariate distributions; conditional distributions
- 309 Bivariate distributions; conditional distributions -- Example 1
- 310 Bivariate distributions; conditional distributions -- Example 2
- 311 Bivariate distribution summary
- 312 Bivariate random variables independence definition
- 313 Bivariate random variables independence -- Example 1
- 314 Bivariate random variables independence -- Example 2
- 315 Bivariate random variables independence result
- 316 Bivariate random variables independence -- Example 3
- 317 Bivariate random variables independence -- Example 4
- 318 Bivariate random variables expected value definition
- 319 Bivariate random variables expected value -- Example 1
- 320 Bivariate random variables expected value -- Example 2
- 321 Bivariate random variables expected value E[g(X) + h(Y)]
- 322 Bivariate random variables expected value E[g(X) h(Y)]
- 323 Bivariate random variables expected value topics outline
- 324 Covariance definition
- 325 Covariance -- Example 1
- 326 Covariance notes
- 327 Covariance shortcut formula
- 328 Covariance -- Example 2
- 329 Covariance result V[X + Y] = V[X] + V[Y] + 2 Cov(X, Y)
- 330 Covariance and independence
- 331 Correlation definition
- 332 Correlation results
- 333 Correlation lies between -1 and 1
- 334 Correlation -- Example 1
- 335 Conditional expectation definition
- 336 Conditional expectation -- Example 1
- 337 Conditional expectation -- Example 2
- 338 Conditional expectation -- Example 3
- 339 Conditional expectation notes
- 340 Bivariate distributions; moment generating functions
- 341 Bivariate distributions; marginal moment generating functions
- 342 Bivariate normal distribution definition
- 343 Bivariate normal distribution level surfaces
- 344 Bivariate normal distribution with rho = 0
- 345 Bivariate normal distribution marginal distributions
- 346 Bivariate normal distribution conditional distributions
- 347 Bivariate normal distribution homoscedasticity
- 348 Bivariate normal distribution -- Example 1
- 349 Bivariate normal distribution moment generating function
- 350 Bivariate normal distribution -- Example 2
- 351 Bivariate normal distribution matrix approach
- 352 Bivariate normal distribution -- Example 3
- 353 Bivariate normal distribution summary
- 354 Multivariate random variables definition
- 355 Multivariate random variables joint pmf/pdf existence conditions
- 356 Multivariate distributions -- Example 1
- 357 Multivariate distributions -- Example 2
- 358 Multivariate distributions: joint cumulative distribution functions
- 359 Multivariate distributions: joint cumulative distribution functions -- Example 1
- 360 Multivariate distributions: Marginal distributions -- Example 1
- 361 Multivariate distributions: Conditional distributions -- Example 1
- 362 Multivariate distributions: Independence
- 363 Multivariate distributions: Independence -- Example 1
- 364 Multivariate distributions: Independence -- Example 2
- 365 Multinomial distribution
- 366 Multinomial distribution -- Example 1
- 367 Multivariate distributions: Expectation
- 368 Multivariate distributions: Expected value of a sum
- 369 Multivariate distributions: Expected value of a sum -- Example 1
- 370 Multivariate distributions: Expected value of a sum -- Example 2
- 371 Multivariate distributions: Expected value of a product
- 372 Multivariate distributions: Variance of a sum of random variables
- 373 Multivariate distributions: Variance of a sum of random variables -- Example 1
- 374 Multivariate distributions: Variance of a sum of random variables -- Example 2
- 375 Multivariate distributions: Variance of a sum of random variables -- Example 3
- 376 Multivariate distributions: Joint moment generating functions
- 377 Multivariate distributions: Matrix representation -- Example 1
- 378 Multivariate distributions: Matrix representation -- Example 2
- 379 Multivariate normal distribution
- 380 Multivariate normal distribution results
- 381 Functions of random variables
- 382 Function definition
- 383 Functions -- Example 1
- 384 Functions: One-to-one
- 385 Functions -- Example 2
- 386 Functions -- Example 3
- 387 Functions: Other varieties
- 388 Cumulative distribution function technique
- 389 Chapter 7 roadmap
- 390 Cumulative distribution technique -- Example 1
- 391 Cumulative distribution technique -- Example 2
- 392 Cumulative distribution technique -- Example 3
- 393 Cumulative distribution technique -- Example 4
- 394 Transformation technique for discrete random variables
- 395 Transformation technique for discrete random variables -- Example 1
- 396 Transformation technique for continuous random variables
- 397 Transformation technique for continuous random variables -- Example 1
- 398 Transformation technique for bivariate discrete random variables
- 399 Transformation technique for bivariate discrete random variables -- Example 1
- 400 Transformation technique for bivariate continuous random variables
- 401 Transformation technique for bivariate continuous random variables -- Example 1
- 402 Transformation technique for bivariate continuous random variables -- Example 2
- 403 Transformation technique for bivariate continuous random variables -- Example 3
- 404 Order statistics
- 405 Order statistics -- Example 1
- 406 Order statistics -- Example 2
- 407 Order statistics joint distribution result
- 408 Order Statistics -- Example 3
- 409 Order statistics -- Example 4
- 410 Order statistics marginal distributions result
- 411 Order statistics special cases
- 412 Order statistics -- Example 5
- 413 Order statistics -- Example 6
- 414 Order statistics -- Example 7
- 415 Moment generating function technique
- 416 Moment generating function technique -- Example 1
- 417 Moment generating function technique -- Example 2
