The following distributions arise on the basis of assumptions similar to those leading to (1.1)–(1.3).

(i) Independent trials with constant probability p of success are carried out until a preassigned number m of successes has been obtained. If the number of trials required is X + m, then X has the negative binomial distribution Nb(p, m):
P{X = x} =

m + x − 1 x pm(1 − p)
x, x = 0, 1, 2 ....
(ii) In a sequence of random events, the number of events occurring in any time interval of length τ has the Poisson distribution P(λτ ), and the numbers of events in nonoverlapping time intervals are independent. Then the “waiting time” T, which elapses from the starting point, say t = 0, until the first event occurs, has the exponential probability density p(t) = λe−λτ , t ≥ 0.
Let Ti, i ≥ 2, be the time elapsing from the occurrence of the (i − 1)st event to that of the ith event. Then it is also true, although more difficult to prove, that T1, T2,... are identically and independently distributed. A proof is given, for example, in Karlin and Taylor (1975).
(iii) A point X is selected “at random” in the interval

(a, b), that is, the probability of X falling in any subinterval of

(a,

b) depends only on the length of the subinterval, not on its position. Then X has the uniform distribution U

(a,

b) with probability density p(x)=1/(b − a), a < x < b.
Section 1.5