Using+Distributions

Using Distributions (Distribution Theory) (GWU EMSE-271)
Index | Topics (Logical Lectures) | Lectures | Problems | Readings | Nomenclature | Concepts

**PLACEHOLDER:** Based on my previous experience, more on this topic would be useful. See Mid Term study notes email.

Van Dorp Topics:
 * Discrete Univariate Models (Slide 46)
 * Continuous Univariate Models (Slide 47)
 * Kernals and Constant of Integration (Slide 48) (Parametric Family EMSE 280)


 * In Hypothesis Testing: **
 * **Distribution** || **Reference** || **Comment** || **Used For (Dig up PAD 201 Chart - check) ** ||
 * **Normal** || [|wikipedia] || Can approximately describe "any variable that tends to cluster about the mean." || "Variable that is sum of a large number of independent factors." ||
 * **(Students) t-distribution** || [|wikipedia] || "population standard deviation is unknown and has to be estimated from the data." OR "when the sample size is small." || "popular Student's //t//-tests for the statistical significance of the difference between two sample means, and for confidence intervals for the difference between two population means." ||
 * **Chi-squared** || wikipedia || "easily calculated quantities can be proven to have distributions that approximate to the chi-square distribution if the null hypothesis is true." || "common chi-square tests for goodness of fit of an observed distribution to a theoretical one, and of the independence of two criteria of classification of qualitative data." ||
 * **F Distribution** || [|wikipedia] || "The //F//-distribution arises frequently as the null distribution of a test statistic" || "in likelihood-ratio tests, perhaps most notably in the analysis of variance;" ||

**General Uses:**
 * **Distribution** || **Reference** || **Comment** || **Used For** ||
 * Bernoulli || [|Wikipedia] || Related to binominal distribution. Beta distribution is the conjugate prior. || Coin toss. [|From Wiikipedia] Bernoulli trial [Was the newborn child a girl? Did a citizen vote for a specific candidate? Did a mosquito die after the area was sprayed with insectide?] ||
 * Binomial || [|Wikipedia] || When n= 1 the binomial distribution is a Bernoulli distribution. Also used for approximating hypergeometric distribution (sampling without replacement) || Finite sequence of events (counting number of successes or # of coin tosses in a finite # of trials). From [|Wikipedia] ["number of successes in a sequence of //n// independent yes/no experiments, each of which yields success with probability //p."// Roll of a standard die ten times and count the number of sixes. Flip a coin 3 times and cound the number of heads.] ||
 * Hyper geometric || [|Wikipedia] || HG Test is "one-tailed version of Fischer's exact test." -[|Wikipedia] Related to Bernoulli, binomial and normal distirbutions. - [|Wikipedia] || Distribution of colored balls – What’s the probability of a specific color being pulled? (Urn problem). From Wikipedia ["number of successes in a sequence of //n// draws from a finite population //without// replacement." ||
 * Negative Binomial || Mazzuchi 208 || Fixed number of trials. RV X = # of trials. measured in S or F. Prob is constant for each trial. || Continues until fixed success (or failures) occur – so the last trial will always be a success (or failure, depending). Common example is selling candy bars, your last trial is always a success (you are out of candy bars) ||
 * Poisson || [|Wikipedia] || "The Poisson distribution can be derived as a limiting case of the binomial distribution." - Wikipedia || Arrival process for a bank or call center. From Wikipedia ["probability of a number of events occurring in a fixed period of time if these events occur with a known average rate and independently of the time since the last event" ||
 * Geometric || Wikipedia || || Testing until failure or successful ||
 * Uniform || Wikipedia || || ||
 * Exponential || Wikipedia || Gamma, Weibull, Rayleigh, Gumbel, Laplace, Uniform, Chi-Squared - Wikipedia || Describes the times between events in a Poisson process (independent, constant rate). ||
 * Gamma || Wikipedia || Exponential, chi-squared, ... || Probability model for waiting times, i.e, waiting time until death. ||
 * Normal || Wikipedia || || Height of women or men ||
 * Beta || [|Wikipedia] || Binomial distribution, uniform distribution. Note: Used extensively in Bayesian statistics. || Uncertainties about probabilities ||
 * Chi-squared ||  ||   ||   ||
 * Weibull || Wikipedia ||  ||   ||


 * Sources:**
 * EMSE 271, Fall 2009
 * Wikipeida