1963/01/01 by G. H. Jowett · 1 citation
Computer Science · Economics, Econometrics and Finance · Mathematics · #Bayesian Methods and Mixture Models #Beta-binomial distribution #Binomial (polynomial) #Binomial distribution #Econometrics #Financial Risk and Volatility Modeling #Mathematics #Negative binomial distribution #Negative multinomial distribution #Poisson distribution #Statistical Distribution Estimation and Applications #Statistics
paper · doi:10.2307/2986663
openalex publication_date 1963/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/07
for the binomial distribution is an incomplete Beta integral. The cumulative probability function for the F distribution is also an incomplete Beta integral. The purposes of this note are to demonstrate that this is more than a mathematical coincidence and to show how the standard tables of Fmay be used to evaluate partial binomial sums. 2. Suppose that random events are occurring at a long-run rate of one per unit of time, and that each event independently has a probability p of being of one type, called E-events, and a probability q( = 1 -p) of being of a second type, called E-events. Then in any set of n successive events the number r of E-events will have a binomial distribution, the probability that r is at least equal to any specified value ro being given by (1). Furthermore, the sequence of E-events may itself be regarded as a sequence of random events occurring at a long-run rate of p per unit time, and the sequence of E-events as an independent sequence of random events occurring at a long-run rate of q per unit time; this is a consequence of the randomness of the original sequence coupled with independent occurrence of type at each event. When random events are occurring, the time interval from an arbitrary instant to the instant of occurrence of one such event (or equally between the instants of occurrence of two such events) is an exponentially distributed variate, the mean of the distribution being the reciprocal of the rate of occurrence (v. Jowett (1958)). Since the X22 distribution is in fact an exponential distribution with mean 2, it follows from the additive property of the x2 distribution and the independence of the sequences of E-events and E-events that the time interval T(m) from an arbitrary instant to the mth E-event is distributed as (x2m2)/2p, and the time interval T'(m) to the mth E-event is independently distributed as X2m2/2q. If r > ro, by the time that the nth event has happened, at least 55