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Probability and statistics
Move from uncertain events to random variables, probability models and evidence-based statistical inference.
Events and probability
Probability
A probability measure assigns nonnegative weights to events, with total weight one and countable additivity on disjoint events.
No reviewed resource yetConditional probability
When P(B)>0, conditional probability restricts attention to B and renormalizes the probabilities.
No reviewed resource yetIndependence
Two events are independent when their joint probability equals the product of their probabilities; this differs from being disjoint.
No reviewed resource yetBayes theorem
Bayes theorem relates conditional probabilities in opposite directions using prior probabilities and a nonzero evidence probability.
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Random variables
Random variables
A random variable is a measurable function on a probability space, allowing outcomes to be described numerically.
No reviewed resource yetDistributions
A distribution records how probability is assigned to the values of a random variable, using masses, densities or a cumulative distribution function.
1 reviewed resourcesExpectation
Expectation is a probability-weighted average when defined; not every random variable has a finite expectation.
No reviewed resource yetVariance
Variance measures expected squared deviation from the mean when the relevant moments exist.
No reviewed resource yetNormal distribution
A normal distribution is specified by its mean and positive variance, with a symmetric bell-shaped density.
No reviewed resource yetBinomial distribution
A binomial variable counts successes in a fixed number of independent Bernoulli trials with a common success probability.
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Limits and samples
Law of large numbers
Under suitable assumptions, sample averages approach the population expectation in a specified mode of convergence.
No reviewed resource yetCentral limit theorem
For independent identically distributed variables with finite positive variance, standardized sums approach a normal distribution.
No reviewed resource yetSampling
Sampling selects observations from a population; the design determines what inferences the observed sample can support.
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Statistical inference
Parameter estimation
An estimator uses sample data to estimate a model parameter; bias, variance and consistency describe different properties.
No reviewed resource yetConfidence intervals
A confidence procedure has a repeated-sampling coverage rate; the realized interval is not a posterior probability statement about a fixed parameter.
No reviewed resource yetHypothesis testing
A hypothesis test compares data to a null model while controlling a specified error rate; a p-value is not the probability that the null is true.
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