Some minor typos

lectures/python_by_example.md

@@ -471,21 +471,21 @@ import numpy as np
 import matplotlib.pyplot as plt
 ```
 
-Set $T=200$ and $\alpha = 0.9$.
+Set $T=200 \,$ and $\alpha = 0.9$.
 
 ### Exercise 2
 
-Starting with your solution to exercise 2, plot three simulated time series,
-one for each of the cases $\alpha=0$, $\alpha=0.8$ and $\alpha=0.98$.
+Starting with your solution to exercise 1, plot three simulated time series,
+one for each of the cases: $\alpha=0$, $\alpha=0.8 \,$ and $\alpha=0.98$.
 
 Use a `for` loop to step through the $\alpha$ values.
 
 If you can, add a legend, to help distinguish between the three time series.
 
 Hints:
 
-* If you call the `plot()` function multiple times before calling `show()`, all of the lines you produce will end up on the same figure.
-* For the legend, noted that the expression `'foo' + str(42)` evaluates to `'foo42'`.
+* If you call a `plot()` function multiple times before calling a `show()`, all of the lines you produce will end up on the same figure.
+* For the legend, note that the expression `'foo' + str(42)` evaluates to `'foo42'`.
 
 ### Exercise 3
 
@@ -498,7 +498,7 @@ x_0 = 0
 \quad \text{and} \quad t = 0,\ldots,T
 $$
 
-Use $T=200$, $\alpha = 0.9$ and $\{\epsilon_t\}$ as before.
+Use $T=200$, $\alpha = 0.9 \, $ and $\{\epsilon_t\}$ as before.
 
 Search online for a function that can be used to compute the absolute value $|x_t|$.
 
@@ -578,7 +578,7 @@ for α in α_values:
     x[0] = 0
     for t in range(T):
         x[t+1] = α * x[t] + np.random.randn()
-    plt.plot(x, label=f'$\\alpha = {α}$')
+    plt.plot((x, label = 'α = ' + str(α)) or we can also use plt.plot(x, label=f'$\\alpha = {α}$')
 
 plt.legend()
 plt.show()