- #1

omg!

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1. there are [tex]n[/tex] discrete time steps

2. between every time step, transitions 0 -> 1, 1-> 0 can happen between states denoted by 0 and 1. the two possible transition directions are poisson processes, i.e. number of time steps between transitions

**of the same kind**are exponentially distributed.

3. additionally, there is another unidirectional transition from 1 to 2.

alternatively, you can think of the sequence [tex]X_1, X_2, \ldots, X_n[/tex] of random variables, with [tex]X_i\in\left{0,1,2}\right}[/tex] with the conditions [tex]P[X_{i+1}=1|X_{i}=0]=\text{const.}[/tex], [tex]P[X_{i+1}=0|X_{i}=1]=const.[/tex], [tex]P[X_{i+1}=2|X_{i}=1]=const.[/tex], [tex]P[X_{i+1}=2|X_{i}=0]=0[/tex] and [tex]P[X_{i+1}=0,1|X_{i}=2]=0[/tex].

now take the continuum limit [tex]n\rightarrow\infty[/tex]

the problem is: what is the PDF of time where the state was last 1, i.e. [tex]X(t)=1,X(s)\neq 1,\quad\forall s\geq t[/tex]