some small changes

inputs/lecture_22.tex

@@ -41,7 +41,7 @@
      \end{question}
      We have
      \begin{IEEEeqnarray*}{rCl}
-         \bP[X_1 = 0, X_2 = 0, X_3 = 1]
+         &&\bP[X_1 = 0, X_2 = 0, X_3 = 1]\\
          &=& \bP[X_3 = 0 | X_2 = 0, X_1 = 0] \bP[X_2 = 0, X_1 = 0]\\
          &=& \bP[X_3 = 0 | X_2 = 0] \bP[X_2 = 0, X_1 = 0]\\
          &=& \bP[X_3 = 0 | X_2 = 0] \bP[X_2 = 0 |  X_1 = 0] \bP[X_1 = 0]\\
@@ -63,7 +63,8 @@
      More generally, consider a Matrix $P \in  (0,1)^{n \times n}$
      whose rows sum up to $1$.
      Then we get a Markov Chain with $n$ states
-     by defining $\bP[X_{n+1} = i | X_{n} = j] = P_{i,j}$.
+     by defining
+     \[\bP[X_{n+1} = i | X_{n} = j] = P_{i,j}.\]
  \end{example}
 
 \begin{definition}
@@ -89,8 +90,10 @@
         \item $\bP[X_0 = i] = \alpha(i)$
             for all $i \in E$,
 
-        \item $\bP[X_{n+1} = i_{n+1} | X_0 = i_0, X_1 = i_1, \ldots, X_{n} = i_{n}]
-            = \bP[X_{n+1} = i_{n+1} | X_n = i_n]$
+        \item \begin{IEEEeqnarray*}{rCl}
+            &&\bP[X_{n+1} = i_{n+1} | X_0 = i_0, X_1 = i_1, \ldots, X_{n} = i_{n}]\\
+            &=& \bP[X_{n+1} = i_{n+1} | X_n = i_n]
+            \end{IEEEeqnarray*}
             for all $n = 0, \ldots$, $i_0,\ldots, i_{n+1} \in E$
             (provided $\bP[X_0 = i_0, X_1 = i_1, \ldots, X_n = i_n] \neq 0$ ).
     \end{enumerate}