Math box Approximating the probability of a coalescent event with the exponential distribution

The series of failures (non-coalescence) until a success (coalescence) in the genealogical process can be modeled where time is continuous (a real number) rather than discrete (an integer). The exponential distribution describes situations in which an object initially in one state can change to an alternative state with some probability that remains constant through time. The exponential distribution could be applied to the time until one of many light bulbs fails, for example, as well as to the time until a coalescent event in a population of lineages (see Fig. 3.24). The exponential distribution is described by

Probability of change = ae~at (3.67)

where a is the constant probability of changing states in a time interval of one, t is time, and e is the mathematical constant base of the natural logarithm (e = 2.71828 . . .). The exponential distribution has a mean of is —. That means that the average time to 2N

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