Continuous Probability for a Single Event

Time is a continuous variable so in general one needs a continuous probability distribution to determine the probability that some event will occur prior to a given time if an infinitesimal rate, r, for the probability is known. It can also be shown that the probability that a single event will occur in a time interval, dt, at time t is dP=Q r dt where Q=1-P is the probability that it will not occur.


A plot helps visualize the continuous distribution (solid red curve) and its relation to the infinitesimal rate. In the plot it was assumed that r=0.5 (per unit of time). The infinitesimal rate is just the slope of the continuous probability distribution at t=0 (dotted blue line). This is the justification for using Δp = r Δt as an estimate of the probability for a small interval dt. The probability (dP) that an event will occur after a long period of time decreases because it probably already has happened. The distribution is determined by the infinitesimal rate and the average time for an occurance is 1/r. It should be noted that this is the probability that just one event will occur before time t (or in a time interval of width t).