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**Additional info for Optimal Reliability Modeling : Principles and Applications**

**Example text**

The probability of success in each trial is a constant p. We are interested in the number of trials needed to observe the kth success, denoted by X . For X to take a value of x, x = k, k + 1, k + 2, . . , we must observe k − 1 successes in the ﬁrst x − 1 trials, and a success must be realized in the xth trial. Thus, the pmf of X is Pr(X = x) = x −1 k p (1 − p)x−k , k−1 x = k, k + 1, k + 2, . . 49) A random variable that has such a pmf is said to have a negative binomial distribution with the parameters k and p.

59) The hypergeometric distribution has applications in sampling without replacement. Items are often selected and checked one by one and the checked ones are not placed back into the population. It is especially useful when N is small. This distribution has to be used in this case because the probability of selecting a success item in a certain trial depends on the outcomes of the previous trials. If many success items have been removed from the population already, the probability of getting another success item in the next trial is smaller than in previous trials.

The numbers of events in disjoint intervals of the same length always have the same distribution with the same parameter values. 2 A counting process {N (t), t ≥ 0} is called a homogeneous Poisson process (HPP) with the parameter λ > 0 if the following conditions are satisﬁed: 1. N (0) = 0, 2. the process has stationary and independent increments, 3. the probability for an event to occur in the interval (t, t + t] may be written as λ t + o( t),1 and 4. the probability for more than one event to occur in the interval (t, t + t] is negligible for small t; in other words, this probability can be written as o( t).