Let  be iid random variables having a gamma distribution  where ais known while bis an unknown parameter. Certain probability inequalities are found for  crossing certain boundaries, which lead to confidence sequences for the mean ab. Using an integrated mixture of a certain supermartingale, an inequality is obtained and used to construct a one-sided confidence sequence for b. An obvious special case is when  which complements sequential confidence intervals for the mean bof an exponential distribution. An interesting application is when  are iid  random variables, and based on the usual sample variance, we obtain a confidence sequence for  A two-stage procedure for a fixed-width interval is also discussed, which is compared with the method of Graybill and Connell [4].

confidence sequence, exponential, fixed-width, gamma distribution, inequality, martingale, mean, normal, supermartingale.