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We are often interested in the waiting time distribution f(t) of the time to an event or, similarly, the probability of an individual surviving to a particular time. In applications in following chapters, I am interested in the amount of time it takes a fish to pass a dam after it has reached it, and I consider the effect of adding mortality to the travel time model.
Waiting time distributions (or survival curves) are often formulated in terms of a hazard function, 
(t), which is the instantaneous failure rate at time t given survival through t. More precisely,
(2.18)
(Kalbfleisch and Prentiss, 1980). If we define 
as 1-F(t) (where F(t) is the cumulative distribution function (cdf) of f(t)), then
(2.19)
(Ross, 1983). The hazard function uniquely determines F(t):
. (2.20)
Also,
. (2.21)
The simplest case is when the hazard function is a constant, i.e. (t) = , and the waiting time probability density function is an exponential distribution
. (2.22)
This is equivalent to stating that a Poisson process with rate governs the waiting time to the next event (Ross, 1993).
The case where (t) is not a constant is referred to as a nonhomogeneous Poisson process (Ross, 1993). Define the mean value function as
, (2.23)
and it can be shown that
, (2.24)
and
. (2.25)
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Spatial and Temporal Models of Migrating Juvenile Salmon with Applications.
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