Each Distribution for Random Variables Has:

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Presentation transcript:

Each Distribution for Random Variables Has: Definition Parameters Density or Mass function Cumulative function Range of valid values Mean and Variance

Binomial Distribution Bernoulli trial S = {success, failure} = {0, 1} trials are independent P(success) = constant = p (independent is similar to sampling with replacement) Binomial Distribution n = Bernoulli trials with p (1-p = q) xi = number of success in n trials ~ b(n, p) xi = 0, 1, …., n V(X) = np(1-p) IME 312

Hypergeometric Distribution n = number of items in sample taken (This is similar to Binomial, but sampling without replacement) n = number of items in sample taken N = number of items in sample space r = number of success in sample space of N xi = number of success in n sample taken ~ h(N, n, r) Relation to Binomial Distribution IME 312

Geometric Distribution p = P(success of independent Bernoulli trials) = constant xi = number of trials up to and including the first success for xi = 1, 2, 3, 4, ……. V(X) = (1-p)/p2 IME 312

Negative Binomial Distribution p = P(success of independent Bernoulli trials) = constant xi = number of trials up to and including the r’th success for xi = r’, r’+1, r’+2, r’+3, r’+4, ……. V(X) = r’(1-p)/p2 IME 312, updated Oct 2012

Multinomial Distribution n = number of independent trials k = possible types of outcome for each trial xi = number of outcomes from type i i=1 to k pi = constant probability of having type i outcome So that x1+x2+ …. +xk=n and p1+p2+….+pk=1 IME 312, updated Oct 2012

xi = 0, 1, 2, …... Poisson Distribution P(x > 1 in subinterval) = 0 P(x = 1 in subinterval) = fix and proportional to the length Independent count in each subinterval = mean number of counts in the unit interval > 0 x = number of counts in the unit interval Unit Matching between x and ! Approximation of Binomial by Poisson: xi = 0, 1, 2, …... IME 312