The proof of Gaussian distribution and two-item distribution belong to exponential distribution family

1, what is the index distribution family 1.1 basic description

Exponential distribution is a kind of important distribution family, in statistical inference, exponential distribution family occupies an important position, which is widely used in various fields. Many of the statistical distributions are exponential distribution, with a certain commonality between them, in the study of its statistical properties and distribution characteristics, using the characteristics of the exponential distribution family, the characteristics of this family distribution can be expressed separately. In the statistical inference of generalized linear models, it is often assumed that samples are subjected to exponential distributions.

1.2 Define the exponential distribution family can be written as follows: Here, the η is called the distribution of the natural parameters, a (η) is called the cumulative parent function (also known as the log Partition function). Exp (-α (η)) This amount is the normalized constant of the distribution P (y;η), which is used to ensure that the distribution P (y;η) is integral to Y of 1. T (y) is called full statistics (sufficient statistic), and for the distributions we consider, it is generally considered that T (y) =y. A set of determined T,a and B defines such a distribution family with the η parameter. For different η, we can get the different distributions in the exponential distribution family. 1.3 Mathematical characteristics for the single-parameter exponential distribution of random variables, in mind, respectively, the function of the η-a pair of η for the one or two-order derivative, there are the following conclusions:

1. Expectation of exponential distributed stochastic variables
2. Variance of exponential distributed random variables

2, the Gaussian distribution belongs to the exponential distribution family of the proof for the Gaussian distribution, when the variance is known, (variance has no effect on the parameters of the model, so we can arbitrarily choose a variance), here we make, then its distribution can be expressed as:         in order to move it closer to the exponential distribution group, we make the following representations:                    &N Bsp                            This shows that the Gaussian distribution can be written in the form of an exponential distribution family, so The Gaussian distribution belongs to the exponential distribution family.         Further, we use the nature of the exponential distribution family to verify that there is:                                                    &NBSP ;                                 &NBSP ;   is exactly the expected and variance of the Gaussian distribution, so validation succeeds.   3, two distributions belonging to the exponential distribution family         for a two-item distribution (Bernoulli distribution), each parameter that takes a different mean is φ, which uniquely determines the distribution of a y that belongs to {0,1}. So can be represented as                                 &NB Sp          So the distribution function of the two-item distribution only takes φ as the parameter, which means two items of distribution:                                               &NBS P       In this way, the natural parameters are:, flip it, have:        to further move the two-item distribution to the exponential distribution group, we can make the following representation:      & nbsp                          ,         &NB Sp              This shows that two distributions can be written in the form of an exponential distribution family, so two distributions belong to the exponential distribution family.          Further, we use the nature of the exponential distribution family to verify that there is:                  and nbsp                                  &NBS P                              is just the two-item distribution of expectations and Variance, therefore satisfies the nature.

Gaussian distributions and two distributions are proof of exponential distribution families