N

for two lineages where t is time in generations. Notice that the constant in the exponential expression (equivalent to the a in equation 3.67)

Therefore, the exponential distribution approximates the probability of non-coalescence at each time t.

To determine the chances that a coalescence event occurred at or before some time, call it TC < twhere TC is the time to coalescence and t is time scaled in units of 2N generations, the cumulative exponential distribution is used. The cumulative distribution

effectively adds up the probabilities of coalescence (as one minus the chance of non-coalescence) as time increases to express the probability of a time interval passing without experiencing a coalescent event. The cumulative distribution approaches one as time increases since the chance of non-coalescence decreases toward zero with increasing time. The probability of coalescence increases more rapidly toward one for larger numbers of lineages k. This cumulative distribution is used to determine the waiting times needed to construct coalescent trees.

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