Merge branch 'main' into tom_july

lectures/lln_clt.md

@@ -51,6 +51,9 @@ will converge to their population means.
 
 Let's see an example of the LLN in action before we go further.
 
+```{prf:example}
+:label: lln_ex_ber
+
 Consider a [Bernoulli random variable](https://en.wikipedia.org/wiki/Bernoulli_distribution) $X$ with parameter $p$.
 
 This means that $X$ takes values in $\{0,1\}$ and $\mathbb P\{X=1\} = p$.
@@ -68,6 +71,7 @@ $$
     \mathbb E X 
     = 0 \cdot \mathbb P\{X=0\} + 1 \cdot \mathbb P\{X=1\} = \mathbb P\{X=1\} = p
 $$
+```
 
 We can generate a draw of $X$ with `scipy.stats` (imported as `st`) as follows:
 
@@ -369,7 +373,8 @@ The LLN fails to hold here because the assumption $\mathbb E|X| < \infty$ is vio
 
 The LLN can also fail to hold when the IID assumption is violated.
 
-For example, suppose that
+```{prf:example}
+:label: lln_ex_fail
 
 $$
     X_0 \sim N(0,1)
@@ -384,6 +389,7 @@ $$
 $$
 
 Therefore, the distribution of $\bar X_n$ is $N(0,1)$ for all $n$!
+```
 
 Does this contradict the LLN, which says that the distribution of $\bar X_n$
 collapses to the single point $\mu$?
@@ -439,9 +445,9 @@ n \to \infty
 Here $\stackrel { d } {\to} N(0, \sigma^2)$ indicates [convergence in distribution](https://en.wikipedia.org/wiki/Convergence_of_random_variables#Convergence_in_distribution) to a centered (i.e., zero mean) normal with standard deviation $\sigma$.
 
 
-The striking implication of the CLT is that for **any** distribution with
+The striking implication of the CLT is that for any distribution with
 finite [second moment](https://en.wikipedia.org/wiki/Moment_(mathematics)), the simple operation of adding independent
-copies **always** leads to a Gaussian(Normal) curve.
+copies always leads to a Gaussian(Normal) curve.
 
 
 
@@ -599,7 +605,7 @@ $$
 $$
 
 where $\alpha, \beta, \sigma$ are constants and $\epsilon_1, \epsilon_2,
-\ldots$ is IID and standard norma.
+\ldots$ are IID and standard normal.
 
 Suppose that
 

lectures/unpleasant.md

@@ -97,7 +97,7 @@ $\widetilde R \check B_{-1}$ is a *real* quantity, being measured in time $0$ go
 
 ### Open market operations
 
-At time $0$, government can rearrange its portolio of debts with subject to the following constraint (on open-market operations):
+At time $0$, government can rearrange its portfolio of debts subject to the following constraint (on open-market operations):
 
 $$
 \widetilde R B_{-1} + \frac{m_0}{p_0} = \widetilde R \check B_{-1} + \frac{\check m_0}{p_0}
@@ -152,7 +152,7 @@ running monetary and fiscal policies.
 
 Here, by **fiscal policy** we mean the collection of actions that determine a sequence of net-of-interest government deficits $\{g_t\}_{t=0}^\infty$ that must be financed by issuing to the public  either money or interest bearing bonds.
 
-By **monetary policy** or **debt-management policy**, we  mean the collection of actions that determine how the government divides its  portolio of debts to the public  between interest-bearing parts (government bonds) and non-interest-bearing parts (money).
+By **monetary policy** or **debt-management policy**, we  mean the collection of actions that determine how the government divides its  portfolio of debts to the public  between interest-bearing parts (government bonds) and non-interest-bearing parts (money).
 
 By an **open market operation**, we mean a government monetary policy action in which the government
 (or its delegate, say, a central bank) either buys  government bonds from the public for newly issued money, or sells  bonds to the public and withdraws the money it receives from public circulation.  
@@ -315,7 +315,7 @@ These parameter settings mean that just before time $0$, the "central bank" sell
 
 That leaves the public with less currency but more government interest-bearing bonds.
 
-Since the public has less currency (it's supply has diminished) it is plausible to anticipate that the price level at time $0$ will be driven downward.
+Since the public has less currency (its supply has diminished) it is plausible to anticipate that the price level at time $0$ will be driven downward.
 
 But that is not the end of the story, because this **open market operation** at time $0$ has consequences for future settings of $m_{t+1}$ and the gross-of-interest government deficit $\bar g_t$. 
 
@@ -418,7 +418,7 @@ plt.xlabel('$m_0$')
 plt.show()
 ```
 
-Now let's write and implement code that let's us experiment with the time $0$ open market operation described earlier.
+Now let's write and implement code that lets us experiment with the time $0$ open market operation described earlier.
 
 ```{code-cell} ipython3
 def simulate(m0, model, length=15, p0_guess=1):