A Markov Chain $(X_n)_n$ has the following transition matrix:
$$P = \begin{bmatrix} 0.1 & 0.3 & 0.6\\ 0 & 0.4 & 0.6\\ 0.3&0.2&0.5 \end{bmatrix}$$ with initial distribution $\alpha = (0.2, 0.3, 0.5)$.
Do the following notations mean different things w.r.t. the above information?
- $P(X_1 = 3|X_0 = 1)$
- $P(X_0 = 3|X_1 = 1)$
If Yes, what is that?
Also what does the following mean in case of Markov Chains?
- $P(X_0 = 3, X_1 = 1)$
- $P(X_1 = 3, X_0 = 1)$
These are standard notations from probability theory. Read a comma as “and” and the vertical bar as “given that.” The latter indicates conditional probability; the former a compound event. In particular, $\Pr(X_0=3\mid X_1=1)$ will generally not have the same value as $\Pr(X_0=3,X_1=1)$.