Tom's fixes of two lectures for Matt, June 28

Author: thomassargent30

lectures/ar1_turningpts.md

@@ -210,6 +210,16 @@ Z_t(Y(\omega)) := \left\{
 \end{array} \right.
 $$ -->
 
+
+
+$$
+Z_t(Y(\omega)) := \left. 
+\begin{cases} 
+\ 1 & \text{if } Y_t(\omega)< Y_{t-1}(\omega)< Y_{t-2}(\omega) \geq Y_{t-3}(\omega) \\
+0 & \text{otherwise}
+\end{cases} 
+$$
+
 Then the random variable **time until the next turning point**  is defined as the following **stopping time** with respect to $Z$:
 
 $$
@@ -236,6 +246,18 @@ T_t(Y(\omega)) := \left\{
 \end{array} \right.
 $$ -->
 
+
+$$
+T_t(Y(\omega)) := \left.
+\begin{cases}
+\ 1 & \text{if } Y_{t-2}(\omega)> Y_{t-1}(\omega) > Y_{t}(\omega) \ \text{and } \ Y_{t}(\omega) < Y_{t+1}(\omega) < Y_{t+2}(\omega) \\
+\ -1 & \text{if } Y_{t-2}(\omega)< Y_{t-1}(\omega) < Y_{t}(\omega) \ \text{and } \ Y_{t}(\omega) > Y_{t+1}(\omega) > Y_{t+2}(\omega) \\
+0 & \text{otherwise}
+\end{cases}
+$$
+
+
+
 Define a **positive turning point today or tomorrow** statistic as 
 
 <!-- $$
@@ -246,6 +268,15 @@ P_t(\omega) := \left\{
 \end{array} \right.
 $$ -->
 
+
+$$
+P_t(\omega) := \left. 
+\begin{cases}
+\ 1 & \text{if } T_t(\omega)=1 \ \text{or} \ T_{t+1}(\omega)=1 \\
+0 & \text{otherwise}
+\end{cases}
+$$
+
 This is designed to express the event
 
 - ``after one or two decrease(s), $Y$ will grow for two consecutive quarters'' 

lectures/prob_matrix.md

@@ -88,7 +88,7 @@ We call this the induced probability distribution of random variable $X$.
 
 Before diving in, we'll say a few words about what probability theory means and how it connects to statistics.
 
-These are topics that are also touched on in these quantecon lectures :XXXXX TOM ADD
+These are topics that are also touched on in the quantecon lectures  <https://python.quantecon.org/prob_meaning.html> and <https://python.quantecon.org/navy_captain.html>.
 
 For much of this lecture we'll be discussing  fixed "population" probabilities. 
 
@@ -1671,9 +1671,7 @@ Copula functions are often used to characterize **dependence** of  random variab
 
 **Discrete marginal distribution**
 
-TOM -- REWRITE OR MAYBE DROP PARTS OF 
-
-If no copula function is given, there could be more than one copulings for two given mariginal distributions.
+As mentioned above,  for two given marginal distributions there can be more than one coupling.
 
 For example, consider two  random variables $X, Y$ with distributions