Completed
Bivariate random variables independence result
Class Central Classrooms beta
YouTube videos curated by Class Central.
Classroom Contents
Probability
Automatically move to the next video in the Classroom when playback concludes
- 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
