BUILT-IN MATHEMATICA SYMBOL

FirstPassageTimeDistribution

FirstPassageTimeDistribution[mproc, f]
represents the distribution of times for the Markov process mproc to pass from the initial state to final states f for the first time.

Details Details

FirstPassageTimeDistribution is also known as first hitting time.
The Markov process mproc can be a DiscreteMarkovProcess or ContinuousMarkovProcess.
The probability for time t in a FirstPassageTimeDistribution is equivalent to Probability[x[t]f_{, 0<<t}x[]fx[0]=i, xmproc], where i is the initial state.
If mproc is already in the target state, FirstPassageTimeDistribution gives a distribution for the mean recurrence time.
If the chain is absorbing and the target states are non-absorbing, FirstPassageTimeDistribution gives a distribution conditional on reaching the target states.
FirstPassageTimeDistribution represents a discrete phase-type distribution for discrete-time Markov processes and a continuous phase-type distribution for continuous-time Markov processes.
FirstPassageTimeDistribution can be used with such functions as Mean, Quantile, PDF, and RandomVariate.