This paper is intended to compare between the Bayesian estimate of a failure rate and the failure rate of a mixed distribution. For the sake of simplification, an exponential distribution and a gamma distribution are adopted as a sampling distribution and its natural conjugate prior distribution. The result shows that both the failure rates are being updated using data and they differ in whether they are functions of unobserved future data or not.

Used as a mixing distribution for an unknown Poisson parameter, the gamma distribution leads to the negative binomial distribution. The hyperparameters of the gamma distribution have their own meanings according to what the Poisson parameter represents. Different sources in the randomness of the Poisson parameter give different interpretations of the negative binomial distribution.

We propose an efficient algorithm to find and update sequential patterns when new transactions are added to an existing database. This method reduces time for scanning the existing and new databases since it uses only transactions that influence the length of sequence. This algorithm outperforms existing algorithm when updated sequential pattern found in the whole database are longer than the patterns in the existing database. Experimental results show the reduction in total execution time.