Probability

Explore a comprehensive course on probability theory, covering fundamental concepts and advanced topics over 28 hours. Learn about random experiments, sample spaces, probability axioms, and various probability distributions. Dive into discrete and continuous random variables, expectation, variance, and moment-generating functions. Study multivariate distributions, functions of random variables, and order statistics. Gain practical experience through numerous examples and applications, including the Monty Hall problem, birthday paradox, and Benford's law. Master essential probability techniques and tools used in statistics, data science, and various scientific fields.

Introduction.
Monty Hall (Let's Make a Deal) Problem.
The birthday problem.
Univariate data set.
Bivariate data set.
Bivariate data set.
Multivariate data set.
Simple linear regression.
Enumeration vs. counting.
Multiplication rule.
Multiplication Rule -- Example 1.
Multiplication Rule -- Example 2.
Multiplication Rule -- Example 3.
Multiplication Rule -- Example 4.
Multiplication Rule -- Example 5.
Multiplication Rule -- Example 6.
Permutations.
Permutations -- Example 1.
Permutations -- Example 2.
Circular Permutations -- Example 1.
Circular Permutations -- Example 2.
Nondistinct permutations.
Nondistinct Permutations -- Example 1.
Nondistinct Permutations -- Example 2.
Nondistinct Permutations -- Example 3.
Nondistinct Permutations -- Example 4.
Combinations.
Combinations -- Example 1.
Combinations -- Example 2.
Combinations -- Example 3.
Combinations -- notes.
Partitioning -- Example 1.
Partitioning -- Example 2.
Partitioning -- Example 3.
Partitioning -- Example 4.
Counting techniques -- unifying example.
Set theory.
Operations on sets.
Venn diagrams -- Example 1.
Venn diagrams -- Example 2.
Set theory -- notes.
Set theory -- application.
Probability introduction.
Random experiments.
Sample spaces.
Sample space classification.
Sample space subsets -- events.
Approaches for calculating probabilities.
Relative frequency approach to estimating probability.
Relative frequency approach (limiting case).
Subjective approach to estimating probability.
Classical approach to calculating probability.
Probability axioms.
Complementary probability.
Probability result concerning subsets.
Probability of unions of events.
Computing probabilities -- Example 1.
Computing probabilities -- Example 2.
Computing probabilities -- Example 3.
Computing probabilities -- Example 4.
Computing probabilities -- Example 5.
Computing probabilities -- Example 6.
Computing probabilities -- Example 7.
Computing probabilities -- Example 8.
Computing probabilities -- Example 9.
Computing probabilities -- Example 10.
Computing probabilities -- Example 11.
Conditional probability -- Example 1.
Conditional probability -- Example 2.
Conditional probability notes.
Conditional probability -- Example 3.
Rule of elimination -- law of total probability.
Conditional probability -- Example 4.
Rule of Bayes.
Rule of Bayes -- Example 1.
Rule of Bayes -- Example 2.
Independence.
Independence -- Example 0.
Independence -- Example 1.
Mutual independence.
Independence in a series system.
Independence in a parallel system.
Independence -- Example 2.
Random variables.
Discrete random variable definition.
Discrete random variables -- Example 1.
Probability mass functions.
Discrete random variables -- Example 2.
Discrete random variables -- Example 3.
Discrete random variables -- Example 4.
Discrete random variables -- Example 5.
Discrete random variables -- Example 6.
Discrete random variables -- Example 7.
Discrete random variables -- Example 8.
Discrete random variable summary.
Continuous random variables introduction.
Probability density functions.
Continuous random variables -- Example 0.
Continuous random variables -- Example 1.
Continuous random variables -- Example 2.
Classifying random variables.
Mixed random variables.
Continuous random variables summary.
Cumulative distribution function definition.
Cumulative distribution function notes.
Cumulative distribution function conversion.
Cumulative distribution functions -- Example 1.
Cumulative distribution functions -- Example 2.
Cumulative distribution functions -- Example 3.
Cumulative distribution functions -- Example 4.
Cumulative distribution functions -- Example 5.
Cumulative distribution function of a mixed random variable.
Cumulative distribution function topics.
Percentiles.
Percentiles -- Example 1.
Percentiles -- Example 2.
Percentiles -- Example 3.
Random variate generation.
Random variate generation -- Example 1.
Random variate generation -- Example 2.
Transformations of random variables.
Transformations of random variables -- Example 1.
Transformations of random variables -- Example 2.
APPL introduction.
APPL -- Example 1.
APPL data structure.
APPL -- Example 2.
APPL -- Example 3.
APPL -- Example 4.
Mixtures.
Mixtures -- Example 1.
Mixtures application.
Continuous mixtures.
Expectation.
Expectation -- Example 1.
Expectation -- Example 2.
Expectation -- Example 3.
Expectation -- Example 4.
Expectation -- Example 5.
Expectation -- Example 6.
Expectation -- Example 7.
Expectation -- Example 8.
Expectation -- Example 9.
Measures of central tendency.
Measures of central tendency -- Example 1.
Population mode definition.
Population mean summary.
Expectation topics.
Expectation of a constant.
Expectation of a constant times a random variable.
Expectation of a function of a random variable -- Example 1.
Expectation of a function of a random variable -- Example 1.
Expectation of the function of a random variable -- Example 1.
Expectation of a constant times a function of a random variable.
Expectation of the sum of two functions of a random variable.
Population variance definition.
Notes on population variance.
Population variance shortcut formula.
Population variance -- Example 1.
Population variance of aX+b.
Population variance corollaries.
Moment definition.
Standardized random variables.
Skewness.
Kurtosis.
Skewness and kurtosis -- Example 1.
Moment generating function definition.
Using moment generating functions to generate moments.
Moment generating functions -- Example 1.
Moment generating functions -- Example 2.
Characteristic functions.
Conditional expectation.
Markov's inequality.
Markov's inequality -- Example 1.
Chebyshev's inequality.
Chebyshev's inequality -- Example 1.
Common discrete distributions.
Bernoulli distribution definition.
Bernoulli trials.
Bernoulli distribution moments.
Bernoulli distribution summary.
Binomial distribution definition.
Binomial distribution notes.
Binomial distribution mean.
Binomial distribution moments.
Binomial distribution shape.
Binomial distribution -- Example 1.
Binomial distribution -- Example 2.
Binomial distribution calculations in R.
Binomial distribution -- Example 3.
Binomial distribution -- Example 4.
Binomial distribution -- Example 5.
Binomial distribution summary.
Geometric distribution definition.
Geometric distribution existence conditions.
Geometric distribution cumulative distribution function.
Geometric distribution memoryless property.
Geometric distribution moment generating function.
Geometric distribution population mean.
Geometric distribution moments.
Geometric distribution -- Example 1.
Geometric distribution definition.
Geometric distribution -- Example 2.
Negative binomial distribution.
Negative binomial moment generating function.
Negative binomial distribution -- Example 1.
Negative binomial distribution.
Negative binomial distribution -- Example 2.
Poisson distribution introduction.
Poisson approximation to the binomial distribution.
Poisson distribution definition.
Poisson distribution moment generating function.
Poisson distribution -- Example 1.
Poisson processes introduction.
Poisson processes illustrations.
Poisson process notation.
Poisson process counting function.
Poisson processes -- Example 1.
Poisson process time between arrivals.
Poisson process superpositioning.
Poisson process decomposition.
Poisson processes and order statistics.
Poisson process summary.
Poisson distribution -- Horse kick data.
Hypergeometric distribution introduction.
Hypergeometric distribution.
Hypergeometric distribution support.
Hypergeometric distribution moments.
Hypergeometric distribution -- Example 1.
Discrete uniform distribution.
Discrete uniform distribution -- Example 1.
Benford's law -- Benford distribution.
Zipf distribution.
Zipf distribution -- Example 1.
Mixture distribution.
Discrete distribution summary.
Common continuous distributions.
Uniform distribution.
Uniform distribution cumulative distribution function.
Uniform distribution moment generating function.
Uniform distribution moments.
Uniform distribution -- Example 1.
Uniform distribution -- Example 2.
Uniform distribution -- Example 3.
Uniform distribution -- Example 4.
Uniform distribution -- Example 5.
Uniform distribution -- Example 6.
Exponential distribution definition.
Exponential distribution rate parameter.
Exponential distribution cumulative distribution function.
Exponential distribution memoryless property.
Exponential distribution moment generating function.
Exponential distribution moments.
Gamma function.
Exponential distribution -- Example 1.
Exponential distribution -- Example 2.
Exponential distribution -- Example 3.
Exponential distribution -- Example 4.
Exponential distribution summary.
Gamma Distribution Definition.
Gamma distribution moment generating function.
Gamma distribution moments.
Gamma distribution special cases.
Gamma distribution -- Example 1.
Gamma distribution summary.
Normal distribution introduction.
Normal distribution history.
Normal distribution properties.
Normal distribution computations.
Normal distribution moment generating function.
Normal distribution Y = a + bX result.
Normal distribution Z = (X - mu) / sigma.
Normal distribution Y = ((X - mu) / sigma) ^ 2 result.
Normal distribution -- Example 1.
Normal distribution -- Example 2.
Other continuous distributions.
Beta distribution.
Beta function.
Beta distribution mean.
Beta distribution moments.
Beta distribution -- Example 1.
Beta distribution -- Example 2.
Triangular distribution.
Triangular distribution cumulative distribution function.
Triangular distribution moments.
Triangular distribution -- Example 1.
Weibull Distribution.
Weibull distribution moments.
Weibull distribution -- Example 1.
Continuous distributions summary.
Multivariate distributions introduction.
Bivariate distribution introduction.
Bivariate distributions -- Automobile illustration.
Bivariate distribution definition.
Bivariate distribution pmf/pdf.
Bivariate distributions -- Example 1.
Bivariate distributions -- Example 2.
Bivariate distributions -- Example 3.
Bivariate distributions -- Example 4.
Bivariate distributions -- Example 5.
Bivariate distributions notation.
Bivariate distributions cumulative distribution functions.
Bivariate distributions cumulative distribution functions -- Example 1.
Bivariate distributions; marginal distributions.
Bivariate distributions; marginal distributions -- Example 1.
Bivariate distributions; marginal distributions -- Example 2.
Bivariate distributions; marginal distributions -- Example 3.
Bivariate distributions; conditional distributions.
Bivariate distributions; conditional distributions -- Example 1.
Bivariate distributions; conditional distributions -- Example 2.
Bivariate distribution summary.
Bivariate random variables independence definition.
Bivariate random variables independence -- Example 1.
Bivariate random variables independence -- Example 2.
Bivariate random variables independence result.
Bivariate random variables independence -- Example 3.
Bivariate random variables independence -- Example 4.
Bivariate random variables expected value definition.
Bivariate random variables expected value -- Example 1.
Bivariate random variables expected value -- Example 2.
Bivariate random variables expected value E[g(X) + h(Y)].
Bivariate random variables expected value E[g(X) h(Y)].
Bivariate random variables expected value topics outline.
Covariance definition.
Covariance -- Example 1.
Covariance notes.
Covariance shortcut formula.
Covariance -- Example 2.
Covariance result V[X + Y] = V[X] + V[Y] + 2 Cov(X, Y).
Covariance and independence.
Correlation definition.
Correlation results.
Correlation lies between -1 and 1.
Correlation -- Example 1.
Conditional expectation definition.
Conditional expectation -- Example 1.
Conditional expectation -- Example 2.
Conditional expectation -- Example 3.
Conditional expectation notes.
Bivariate distributions; moment generating functions.
Bivariate distributions; marginal moment generating functions.
Bivariate normal distribution definition.
Bivariate normal distribution level surfaces.
Bivariate normal distribution with rho = 0.
Bivariate normal distribution marginal distributions.
Bivariate normal distribution conditional distributions.
Bivariate normal distribution homoscedasticity.
Bivariate normal distribution -- Example 1.
Bivariate normal distribution moment generating function.
Bivariate normal distribution -- Example 2.
Bivariate normal distribution matrix approach.
Bivariate normal distribution -- Example 3.
Bivariate normal distribution summary.
Multivariate random variables definition.
Multivariate random variables joint pmf/pdf existence conditions.
Multivariate distributions -- Example 1.
Multivariate distributions -- Example 2.
Multivariate distributions: joint cumulative distribution functions.
Multivariate distributions: joint cumulative distribution functions -- Example 1.
Multivariate distributions: Marginal distributions -- Example 1.
Multivariate distributions: Conditional distributions -- Example 1.
Multivariate distributions: Independence.
Multivariate distributions: Independence -- Example 1.
Multivariate distributions: Independence -- Example 2.
Multinomial distribution.
Multinomial distribution -- Example 1.
Multivariate distributions: Expectation.
Multivariate distributions: Expected value of a sum.
Multivariate distributions: Expected value of a sum -- Example 1.
Multivariate distributions: Expected value of a sum -- Example 2.
Multivariate distributions: Expected value of a product.
Multivariate distributions: Variance of a sum of random variables.
Multivariate distributions: Variance of a sum of random variables -- Example 1.
Multivariate distributions: Variance of a sum of random variables -- Example 2.
Multivariate distributions: Variance of a sum of random variables -- Example 3.
Multivariate distributions: Joint moment generating functions.
Multivariate distributions: Matrix representation -- Example 1.
Multivariate distributions: Matrix representation -- Example 2.
Multivariate normal distribution.
Multivariate normal distribution results.
Functions of random variables.
Function definition.
Functions -- Example 1.
Functions: One-to-one.
Functions -- Example 2.
Functions -- Example 3.
Functions: Other varieties.
Cumulative distribution function technique.
Chapter 7 roadmap.
Cumulative distribution technique -- Example 1.
Cumulative distribution technique -- Example 2.
Cumulative distribution technique -- Example 3.
Cumulative distribution technique -- Example 4.
Transformation technique for discrete random variables.
Transformation technique for discrete random variables -- Example 1.
Transformation technique for continuous random variables.
Transformation technique for continuous random variables -- Example 1.
Transformation technique for bivariate discrete random variables.
Transformation technique for bivariate discrete random variables -- Example 1.
Transformation technique for bivariate continuous random variables.
Transformation technique for bivariate continuous random variables -- Example 1.
Transformation technique for bivariate continuous random variables -- Example 2.
Transformation technique for bivariate continuous random variables -- Example 3.
Order statistics.
Order statistics -- Example 1.
Order statistics -- Example 2.
Order statistics joint distribution result.
Order Statistics -- Example 3.
Order statistics -- Example 4.
Order statistics marginal distributions result.
Order statistics special cases.
Order statistics -- Example 5.
Order statistics -- Example 6.
Order statistics -- Example 7.
Moment generating function technique.
Moment generating function technique -- Example 1.
Moment generating function technique -- Example 2.