We study the relations between F and G,that is,we obtains the closure and asymptotic behavior for random sums of many heavy-tailed distribution classes on(-∞,+∞),and apply them to the compound Poisson distribution and the compound geometric distribution.

The effect of bi-particle s geometrical distribution is then investigated.

On the assumption that a single server provides service to the customers in the system,and the batch arrival of the customers submits to the geometrical distribution,the number of the customers in a batch arrival is random variables and has the generalized distribute function,and the service time also depends on the geometrical distribution,using the method of embedded the Markov chain,the.

And two strong deviation theorems for the relativeentropy density of geometrical distribution are derived.

The paper derives the solution of the last probability of ruin in geometry distribution model by probability theory,and obtains its approachable estimate.

The limit properties of deviation of relative entropy with respect to the independent geometry distribution are disc ussed and three strong deviation theorems of relative entropy density which give the estimate of strong large number of the deviation when n→∞ are obtained.

For any information source on a countable set, the limit properties of relative likelihood ratio and log-likelihood ratio of entropy density with respect to the independent geometry distribution, an important problem in the information theory is discussed.

The statistical analysis for geometric distribution with full sample size based on tampered failure rate model under step-stress accelerated life testing;

Interval estimations of parameters for geometric distribution and research of statistical approach;

Parameter estimation of geometric distribution under Q-symmetric entropy loss function;

This article tells us how to use probability generating function to find the expectation and variance of super geometry distribution.