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About the history of gamma distribution?
According to the links, Karl Pearson was first to formally introduce the gamma distribution. However, the symbol gamma for the gamma function, as a part of calculus, originated far earlier, by Legrenge (1752 to 1853). The beta and gamma functions are related. Please review the related links, particularly the second one from Wikipedia.
How gamma are members of the exponential family prove?
Think you've got this backwards. The exponential probability distribution is a gamma probability distribution only when the first parameter, k is set to 1. Consistent with the link below, if random variable X is distributed gamma(k,theta), then for gamma(1, theta), the random variable is distributed exponentially. The gamma function in the denominator is equal to 1 when k=1. The denominator will reduce to theta when k = 1. The first term will be X0 = 1. using t to represent theta, we have f(x,t) = 1/t*exp(-x/t) or we can substitute L = 1/t, and write an equivalent function: f(x;L) = L*exp(-L*x) for x > 0 See: http://en.wikipedia.org/wiki/Gamma_distribution [edit] To the untrained eye the question might seem backwards after a quick google search, yet qouting wikipedia lacks deeper insight in to the question: What the question is referring to is a class of functions that factor into the following form: f(y;theta) = s(y)t(theta)exp[a(y)b(theta)] = exp[a(y)b(theta) + c(theta) + d(y)] where a(y), d(y) are functions only reliant on y and where b(theta) and c(theta) are answers only reliant on theta, an unkown parameter. if a(y) = y, the distribution is said to be in "canonical form" and b(theta) is often called the "natural parameter" So taking the gamma density function, where alpha is a known shape parameter and the parameter of interest is beta, the scale parameter. The density function follows as: f(y;beta) = {(beta^alpha)*[y^(alpha - 1)]*exp[-y*beta]}/gamma(alpha) where gamma(alpha) is defined as (alpha - 1)! Hence the gamma-density can be factored as follows: f(y;beta) = {(beta^alpha)*[y^(alpha - 1)]*exp[-y*beta]}/gamma(alpha) =exp[alpha*log(beta) + (alpha-1)*log(y) - y*beta - log[gamma(alpha)] from the above expression, the canonical form follows if: a(y) = y b(theta) = -beta c(theta) = alpha*log(beta) d(y) = (alpha - 1)*log(y) - log[gamma(alpha)] which is sufficient to prove that gamma distributions are part of the exponential family.
What is the major difference between a mathematical model and an econometric model?
Mathematical model is exact in nature.it has Beta zero and Beta one and no stochastic or disturbance variables. Econometric model represents omitted variable, error in measurement and stochastic variables.
Difference between alpha testing and beta testing?
Alpha Testing is always performed at the time of Acceptance Testing when developers test the product and project to check whether it meets the user requirements or not.Beta Testing is always performed at the time when software product and project are marketed.
The most commonly used greek letters in statistics?
I'll give you some common Greek symbols used in statistical analyses. I can't tell you which is the most common one given the enormous task of reviewing every statistics book. The Greek mu for mean, sigma for variance and rho for correlation are probably the first ones that one encounters in statistical analyses. Also, beta for beta distribution, gamma for gamma distribution, chi for chi-squared distribution. Alpha and beta are common as distribution parameters. In derivations, delta is common for differences of variables. Tau is common for a time variable. You will find more information in the related link.
