Forward Mark Proof with airSlate SignNow

Forward mark proof

well what we haven't done yet is we haven't yet proved that this process we define in terms of its infinitesimal increments is actually a poisson process so that's what we're going to do here note that it's obvious that it starts from zero and uh the third point of the definition that the indep that the increments are independent is also clear so we've just got to check that the independent that the increments of the process have the poisson distribution and that's what we're going to do here since the increments are stationary are time homogeneous what we want to show is that we want to show that x at time t is poisson lambda t if we can do that we're done so what it will be convenient to do it will be convenient to write p j t equals the probability that x t equals j and if we can prove that that is a e to the minus lambda t 1 to t to the j over j factorial then we'll be home and dry so that's what we'd like to be able to do so what can we say about p j t well throughout this section we've been looking at small time increments so in fact let's think about what we can show about p j t plus tau for some very small time period 10 so this is saying what's the probability we're at j at time t plus tau now we can get to j at times t plus tau in a few different ways one way is that we could already have been at j at time t and then we didn't move so we could have not moved with probability 1 minus lambda tau plus o tau you'll remember that for the probability of not moving so if we were already at j at time t that would be that on the other hand we might have been at j minus one at time t but then seen an arrival in the little period of time so remember that the probability of seeing an arrival and a little period of time tau is lambda tau plus perhaps some lower order terms and so if we'd been at j minus one that's the chance we'd have seen one arrival and that would have taken us up to j or we could have been at j minus two or j minus three or j minus four or j minus five but all those things have very small probability so they're all little o of tau okay uh we can do two things to rearrange here one is that all the little o of towels can go in the same little hour of towel so the little of town the little out of town the little of town we can move all those together the other thing is i'm going to want to take one of these pj t's over to the...