Based on the definition of a CTMC with rates vi and transition probabilities pij, one can...

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Based on the definition of a CTMC with rates vi and transition probabilities pij, one can simulate a CTMC as follows (see also next week's notebook): From state i, (1) sample the time for the next jump from Exp(vi); (2) sample the new state from (pil,..., Pin). Alternatively, we consider the following: From state i, (1) Sample Exp(qij) for all j such that Iij>0; (2) compute their minimum, as the time for the next jump, (3) sample the new state as the value of j corresponding to this minimum (i.e. the argmin). Show that these two procedures are equivalent (hint: Use properties of Exp r.v. seen in class). Based on the definition of a CTMC with rates vi and transition probabilities pij, one can simulate a CTMC as follows (see also next week's notebook): From state i, (1) sample the time for the next jump from Exp(vi); (2) sample the new state from (pil,..., Pin). Alternatively, we consider the following: From state i, (1) Sample Exp(qij) for all j such that Iij>0; (2) compute their minimum, as the time for the next jump, (3) sample the new state as the value of j corresponding to this minimum (i.e. the argmin). Show that these two procedures are equivalent (hint: Use properties of Exp r.v. seen in class).