The Bounds For The Tail Probabilities Of The Sum Of Random Variables With Exponential Distribution

Exponential distribution and geometric distribution are two common and important distributions in probability distributions.It is of great practical significance to study the statistical properties of exponential distribution and geometric distribution.Janson discussed the estimation of the bound of the tail probability of the unweighted sum of independent random variables with geometric distribution.Based on Janson’s research,Lu further optimized the bound of the tail probability of the unweighted sum of independent random variables with geometric distribution,and obtained a more accurate bound.On the basis of the above research contents,and because geometric distribution is the discretization of exponential distribution,this paper extends the above theoretical research results related to geometric distribution to exponential distribution,and obtains some relevant conclusions about the bounds of the tail probabilities of the sum of random variables with exponential distribution.Its main work is as follows:1.Based on the works of Janson and Lu,we study the tail probability of the unweighted sum of independent random variables with exponential distribution,and give the bound of the tail probability of the unweighted sum of independent random variables with exponential distribution.2.On the basis of the work in the first part,we weight the independent random variables,gave the upper and lower bounds of the tail probability of the weight sum of independent random variables with exponential distribution,and improve upper and lower bounds of the tail probabilities of the weighted sum of independent random variables with exponential distribution.3.The conclusion of tail probability of the unweighted sum of independent random variables with exponential distribution are generalized,and the dependent structure is added between random variables to get the bounds of tail probability of the unweighted sum and weighted sum of negatively orthant dependent random variables with exponential distribution.