Talk:Stochastic matrix/Question 4

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[edit] Stochastic matrix Question 4

\mathbf{p}_k = \mathbf{v}P^k \, .
We want to find the probability that the system is in a given state after a given number of time steps. The set of probabilities for each state after k time steps is given by the probability vector pk. The purpose of the formula is that it gives an expression for the probability vector after k time steps in terms of the initial state vector v and the stochastic matric P - so if we know v and P we can find the probability vector at any subsequent time. The "mathematical induction" part just means that we can derive the general formula for pk by looking at the formulae for p1, p2 etc. and then generalising the pattern that we see to k time steps. Can you see where the formulae that I give above for p1, p2 come from ? Can you see how they lead to a general formula for pk ? Gandalf61 (talk) 09:35, 12 February 2008 (UTC)

I am assuming the formulae \mathbf{p}_k = \mathbf{v}P^k \, that you gave above for p1, p2 came from Summation? If this is true then can the formula be put in Sigma notation format \sum_{i=m}^n x_i = x_m + x_{m+1} + x_{m+2} +\cdots+ x_{n-1} + x_n.  ? --Obsolete.fax (talk) 05:28, 17 February 2008 (UTC)