We consider time-changed Poisson processes, and derive the governing difference–differential equations (DDEs) for these processes. In particular, we consider the time-changed Poisson processes where the time-change is inverse Gaussian, or its hitting time process, and discuss the governing...

It is shown that the negative binomial distribution NB(r,p) may arise out of an identical but dependent geometric sequence. Using a general characterization result for NB(r,p), based on a non-negative integer -valued sequence, we show that NB(2,p) may arise as the distribution of the sum of...

Let [Pi]1,[Pi]2,...,[Pi]k be k populations with [Pi]i being exponential with an unknown location parameter [mu]i and a common but known scale parameter [sigma], i=1,...,k. Suppose independent random samples are drawn from the populations [Pi]1,[Pi]2,...,[Pi]k. Let {Xi1,Xi2,...,Xin} denote the...

Let X1,...,Xp be p independent random observations, where Xi is from the ith discrete population with density of the form ui([theta]i)ti(xi)[theta]ixi, where [theta]i is the positive unknown parameter. Let X(1)=...=X(l)X(l+1)=...=X(m)X(m+1)=...=X(p) denote the ordered observations, where the...

An upper bound for the total variation distance between the distribution of the sum of a sequence of r.v.'s and that of a compound Poisson is derived. Its applications to a general independent sequence and Markov-binomial sequence are demonstrated.