Probability distributions & random samples
Slides are licensed under CC BY-NC-SA 4.0
.Rmd
E.g. the sum of two fair dice: Categorical distribution
Support: Integers from 2 to 12 (discrete)
E.g. the time between shark attacks in Austrailia: Exponential distribution ()
Support: Non-negative real numbers
E.g. questions measuring authoritarian attitudes: Bivariate categorical distribution
Joint probability distributions measure probability across multiple variables and the association between those variables.
Conditional probability measures probability of one variable in a joint distribution, holding the other constant at a specific value. The probabilities must be normalized.
Marginal probability measures probability of one variable in a joint distribution, across all possible values of the other.
(Statisticians have devised and named innumerable distributions over time. See https://en.wikipedia.org/wiki/List_of_probability_distributions for an incomplete list)
Using probability distributions to link (known) data with (unknown) parameters succinctly and clearly communicates a model.
Example from last week: Estimating the unemployment rate p from count of unemployed (Y) in our sample of n individuals
Changes to the model are clear:
Posterior distributions contain a lot of information
Describe the “center” of the distribution Mean, median, and mode all have different meanings
Describe the “spread” of the distribution
Percentile (aka quantile) intervals leave the same amount of density on either end of the distribution.
Highest posterior density intervals find the narrowest possible interval containin the target density.
Figures by Peter McMahan (source code)
Poster detail for Jaws (1975)
Photo by Leo Reynolds on Flickr
Still from Wheel of Fortune