A Markov chain $(ξ_n,n ∈ \mathbb{Z}_+)$ has an initial state $ξ_0 = 0$ and transient probabilities $P(ξ_{n+1} = k+1| ξ_n = k) = p, P(ξ_{n+1} = k| ξ_n = k) = 1 − p, k,n ∈ \mathbb{N}, p ∈ [0,1]$. I find the distribution $ξ_n$.
I do not understand how to calculate the distribution of this process. I had an idea to find a stationary distribution, but since the phase space for a given Markov chain is natural numbers, I cannot write down the transition probability matrix and write down linear relations for stationary distributions. Can anyone help? Thank you!
$\xi_n \in \{0,1,2,\ldots,n\}$. For $k, 0\le k \le n$ we must have $k$ times $ \uparrow$ and $n-k$ times $\to$. We can do this in $C_n^k$ ways.