From 088d92b958b53f5a2719397cdf9c5e71d8e187da Mon Sep 17 00:00:00 2001
From: mmcky <mamckay@gmail.com>
Date: Tue, 16 Jul 2024 08:37:53 +1000
Subject: [PATCH] fix syntax for example

---
 lectures/markov_chains_I.md | 15 +++++----------
 1 file changed, 5 insertions(+), 10 deletions(-)

diff --git a/lectures/markov_chains_I.md b/lectures/markov_chains_I.md
index 2c13cc17..2948239c 100644
--- a/lectures/markov_chains_I.md
+++ b/lectures/markov_chains_I.md
@@ -636,17 +636,15 @@ The probability of being in recession (either mild or severe) in 6 months' time
 $$
 (\psi_t P^6)(1) + (\psi_t P^6)(2)
 $$
-```{index} single: Markov Chains; Cross-Sectional Distributions
+
 ```
 
 ```{index} single: Markov Chains; Cross-Sectional Distributions
 ```
 
-```{prf:example} Cross-Sectional Distributions
+````{prf:example} Cross-Sectional Distributions
 :label: cross-sectional-distributions
 
-### Example 2: Cross-Sectional Distributions
-
 The distributions we have been studying can be viewed either
 
 1. as probabilities or
@@ -674,16 +672,15 @@ So for a very large (tending to infinite) population, $\psi_t P^{10}$ also repre
 
 This is exactly the cross-sectional distribution.
 
- ```{note}
+```{note}
 A cross-sectional frequency measures how a particular variable (e.g., employment status) is distributed across a population at a specific time, providing information on the proportions of individuals in each possible state of that variable.
 ```
 
-
+````
 
 (stationary)=
 ## Stationary distributions
 
-
 As seen in {eq}`fin_mc_fr`, we can shift a distribution forward one
 unit of time via postmultiplication by $P$.
 
@@ -698,8 +695,6 @@ P = np.array([[0.4, 0.6],
 
 Notice that `ψ @ P` is the same as `ψ`.
 
-
-
 Such distributions are called **stationary** or **invariant**.
 
 (mc_stat_dd)=
@@ -753,7 +748,7 @@ corresponds to unemployment (state 0).
 Using $\psi^* = \psi^* P$ and a bit of algebra yields
 
 $$
-    p = \frac{\beta}{\alpha + \beta}
+p = \frac{\beta}{\alpha + \beta}
 $$
 
 This is, in some sense, a steady state probability of unemployment.