[markov_markov_II] Fix prf:theorem in markov_chains_II (#556)

* fix `prf:theorem` in markov_chains_II

* fix a small bug in indexing for prob_dist

Author: HumphreyYang

lectures/markov_chains_II.md

@@ -71,7 +71,7 @@ that
 The stochastic matrix $P$ is called **irreducible** if all states communicate;
 that is, if $x$ and $y$ communicate for all $(x, y)$ in $S \times S$.
 
-```{prf:example}
+````{prf:example}
 :label: mc2_ex_ir
 For example, consider the following transition probabilities for wealth of a
 fictitious set of households
@@ -81,8 +81,6 @@ fictitious set of households
 :align: center
 ```
 
-
-
 We can translate this into a stochastic matrix, putting zeros where
 there's no edge between nodes
 
@@ -97,7 +95,7 @@ $$
 
 It's clear from the graph that this stochastic matrix is irreducible: we can  eventually
 reach any state from any other state.
-```
+````
 
 We can also test this using [QuantEcon.py](http://quantecon.org/quantecon-py)'s MarkovChain class
 
@@ -110,7 +108,7 @@ mc = qe.MarkovChain(P, ('poor', 'middle', 'rich'))
 mc.is_irreducible
 ```
 
-```{prf:example}
+````{prf:example}
 :label: mc2_ex_pf
 
 Here's a more pessimistic scenario in which  poor people remain poor forever
@@ -122,7 +120,7 @@ Here's a more pessimistic scenario in which  poor people remain poor forever
 
 This stochastic matrix is not irreducible since, for example, rich is not
 accessible from poor.
-```
+````
 
 Let's confirm this
 
@@ -279,7 +277,7 @@ In any of these cases, ergodicity will hold.
 
 ### Example: a periodic chain
 
-```{prf:example}
+````{prf:example}
 :label: mc2_ex_pc
 
 Let's look at the following example with states 0 and 1:
@@ -301,7 +299,8 @@ The transition graph shows that this model is irreducible.
 ```
 
 Notice that there is a periodic cycle --- the state cycles between the two states in a regular way.
-```
+````
+
 Not surprisingly, this property 
 is called [periodicity](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/16%3A_Markov_Processes/16.05%3A_Periodicity_of_Discrete-Time_Chains).
 

lectures/prob_dist.md

@@ -882,7 +882,7 @@ x_costco = prices.pct_change()[1:] * 100
 
 ```{code-cell} ipython3
 fig, ax = plt.subplots()
-ax.violinplot([x_amazon, x_costco])
+ax.violinplot([x_amazon['AMZN'], x_costco['COST']])
 ax.set_ylabel('monthly return (percent change)')
 ax.set_xlabel('retailers')