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1、Diagram of distribution relationshipsProbability distributions have a surprising number inter-connections. A dashed line in the chart below indicates an approximate (limit) relationship between two distribution families. A solid line indicates an exact relationship: special case, sum, or transformat
2、ion.Click on a distribution for the parameterization of that distribution. Click on an arrow for details on the relationship represented by the arrow. Other diagrams on this site:The chart above is adapted from the chart originally published by Lawrence Leemis in 1986 (Relationships Among Common Uni
3、variate Distributions, American Statistician 40:143-146.) Leemis published a larger chart in 2008 which is available online.The precise relationships between distributions depend on parameterization. The relationships detailed below depend on the following parameterizations for the PDFs.Let C(n, k)
4、denote the binomial coefficient(n, k) and B(a, b) = (a) (b) / (a + b).Geometric: f(x) = p (1-p)x for non-negative integers x.Discrete uniform: f(x) = 1/n for x = 1, 2, ., n.Negative binomial: f(x) = C(r + x - 1, x) pr(1-p)x for non-negative integers x. See notes on the negative binomial distribution
5、.Beta binomial: f(x) = C(n, x) B( + x, n + - x) / B(, ) for x = 0, 1, ., n.Hypergeometric: f(x) = C(M, x) C(N-M, K - x) / C(N, K) for x = 0, 1, ., N.Poisson: f(x) = exp(-) x/ x! for non-negative integers x. The parameter is both the mean and the variance.Binomial: f(x) = C(n, x) px(1 - p)n-x for x =
6、 0, 1, ., n.Bernoulli: f(x) = px(1 - p)1-x where x = 0 or 1.Lognormal: f(x) = (22)-1/2 exp( -(log(x) - )2/ 22) / x for positive x. Note that and 2are not the mean and variance of the distribution.Normal : f(x) = (2 2)-1/2 exp( - (x - )/)2 ) for all x.Beta: f(x) = ( + ) x-1(1 - x)-1 / () () for 0 x 1
7、.Standard normal: f(x) = (2)-1/2 exp( -x2/2) for all x.Chi-squared: f(x) = x-/2-1 exp(-x/2) / (/2) 2/2 for positive x. The parameter is called the degrees of freedom.Gamma: f(x) = - x-1 exp(-x/) / () for positive x. The parameter is called the shape and is the scale.Uniform: f(x) = 1 for 0 x 1.Cauch
8、y: f(x) = /( (x - )2 + 2) ) for all x. Note that and are location and scale parameters. The Cauchy distribution has no mean or variance.Snedecor F: f(x) is proportional to x(1 - 2)/2 / (1 + (1/2) x)(1 + 2)/2 for positive x.Exponential: f(x) = exp(-x/)/ for positive x. The parameter is the mean.Stude
9、nt t: f(x) is proportional to (1 + (x2/)-( + 1)/2 for positive x. The parameter is called the degrees of freedom.Weibull: f(x) = (/) x-1 exp(- x/) for positive x. The parameter is the shape and is the scale.Double exponential : f(x) = exp(-|x-|/) / 2 for all x. The parameter is the location and mean
10、; is the scale.For comparison, see distribution parameterizations in R/S-PLUS and Mathematica.In all statements about two random variables, the random variables are implicitly independent.Geometric / negative binomial: If each Xi is geometric random variable with probability of success p then the su
11、m of n Xis is a negative binomial random variable with parameters n and p.Negative binomial / geometric: A negative binomial distribution with r = 1 is a geometric distribution.Negative binomial / Poisson: If X has a negative binomial random variable with r large, p near 1, and r(1-p) = , then FX FY
12、 where Y is a Poisson random variable with mean .Beta-binomial / discrete uniform: A beta-binomial (n, 1, 1) random variable is a discrete uniform random variable over the values 0 . n.Beta-binomial / binomial: Let X be a beta-binomial random variable with parameters (n, , ). Let p = /( + ) and supp
13、ose + is large. If Y is a binomial(n, p) random variable then FX FY.Hypergeometric / binomial: The difference between a hypergeometric distribution and a binomial distribution is the difference between sampling without replacement and sampling with replacement. As the population size increases relat
14、ive to the sample size, the difference becomes negligible.Geometric / geometric: If X1 and X2 are geometric random variables with probability of success p1 and p2 respectively, then min(X1, X2) is a geometric random variable with probability of success p = p1 + p2 - p1 p2. The relationship is simple
15、r in terms of failure probabilities: q = q1 q2.Poisson / Poisson: If X1 and X2 are Poisson random variables with means 1 and 2respectively, then X1 + X2 is a Poisson random variable with mean 1 + 2.Binomial / Poisson: If X is a binomial(n, p) random variable and Y is a Poisson(np) distribution then
16、P(X = n) P(Y = n) if n is large and np is small. For more information, see Poisson approximation to binomial.Binomial / Bernoulli: If X is a binomial(n, p) random variable with n = 1, X is a Bernoulli(p) random variable.Bernoulli / Binomial: The sum of n Bernoulli(p) random variables is a binomial(n, p) random variable.Poisson / normal: If X is a
