# Distibution

Distributions are critical in understanding the properties of stochastic variables. A distribution of a number describes the likelihood the variable will taken on a particular value or, in the continuous case, interval.

 Important Terms a) ContinuousThe values of the distribution are express in continuous numbers (1.001, 1.002, .1003, …). b) DiscreteThe values of the distribution are express in whole numbers (1,2,3…). c) MeanMean or average is the central tendency of population. It is the sum of the values divided by the size of the population. d) VarianceThe variance measures of how much the values in the sample differ from the mean on average. e) Standard DeviationThe standard deviation is the square root of the variance f) KurtosisThe measure of peakness of a distribution is Kurosis. g) SkewnessSkewness measures the degree to which a distribution is not symmetric. Common Distrubutions Normal The most familiar distribution, the Normal or Gaussian distribution is used in a variety of test and models.  A key assumption to many models is the normality of the error structure. It is symmetric and continuous. Log NormalThe lognormal distribution is related to the Normal distribution. The logarithm of the values from a lognormal distribution are distributed normally. It is asymmetric and continuous. BernoulliIf there are only two possible outcomes from an experiment then the number of success has a Bernoulli distribution. It is asymmetric and discreet. BinomialUsed in repeated trials with two outcomes, x success in n trails without replacement is pulled from a Binomial distribution.It is symmetric and discreet. MultinomialWhen there are more than two outcomes use a multinomial instead of a binomial Negative BinomialUsed to answer the question number of trials required for kth success to occur.It is asymmetric and discreet. GeometricUsed to answer the question number of trials required for kth success to occur.It is asymmetric and discreet. HypergeometricHypergeometric is a binomial with replacement. It is used to answers the question how many successes in n trials with replacement. PoissonIt is a computationally simpler than the binomial and can be used to answer similar questions. The poisson distribution is used in queuing models for arrival rates. Gamma Gamma is often used when answer questions concerning waiting time between events drawn from a Poisson distribution. Answers question what is the probability of waiting less than x? It is a asymmetric and continuous. ErlangeIt is a special case of the Gamma distribution. The Erlange is also used in queuing model. It is more flexible than the exponential and is used when number of servers is finite. It is a asymmetric and continuous. ExponentialIt is a special case of the Gamma distribution.  It is often used to estimate waiting times in queues.It is a asymmetric and continuous. Chi-SquaredOften used in inferential statistics the Chi-Squared distribution is a special case of the Gama distribution.It is a asymmetric and continuous. Beta Used in Bayesian inference, the Beta distribution can take on a variety of different shapes. It is a asymmetric and continuous. UniformWhen the probability for a given value is constant it has a uniform distribution. It is a symmetric and continuous.