Update functions.md to New API

Author: Chihiro2000GitHub

lectures/functions.md

@@ -281,11 +281,13 @@ We will say more about this {doc}`later <writing_good_code>`.
 Consider again this code from the {doc}`previous lecture <python_by_example>`
 
 ```{code-cell} python3
+rng = np.random.default_rng()
+
 ts_length = 100
 ϵ_values = []   # empty list
 
 for i in range(ts_length):
-    e = np.random.randn()
+    e = rng.standard_normal()
     ϵ_values.append(e)
 
 plt.plot(ϵ_values)
@@ -306,7 +308,7 @@ This is accomplished in the next program
 def generate_data(n):
     ϵ_values = []
     for i in range(n):
-        e = np.random.randn()
+        e = rng.standard_normal()
         ϵ_values.append(e)
     return ϵ_values
 
@@ -336,9 +338,9 @@ def generate_data(n, generator_type):
     ϵ_values = []
     for i in range(n):
         if generator_type == 'U':
-            e = np.random.uniform(0, 1)
+            e = rng.uniform(0, 1)
         else:
-            e = np.random.randn()
+            e = rng.standard_normal()
         ϵ_values.append(e)
     return ϵ_values
 
@@ -371,19 +373,19 @@ def generate_data(n, generator_type):
         ϵ_values.append(e)
     return ϵ_values
 
-data = generate_data(100, np.random.uniform)
+data = generate_data(100, rng.uniform)
 plt.plot(data)
 plt.show()
 ```
 
-Now, when we call the function `generate_data()`, we pass `np.random.uniform`
+Now, when we call the function `generate_data()`, we pass `rng.uniform`
 as the second argument.
 
 This object is a *function*.
 
-When the function call  `generate_data(100, np.random.uniform)` is executed, Python runs the function code block with `n` equal to 100 and the name `generator_type` "bound" to the function `np.random.uniform`.
+When the function call  `generate_data(100, rng.uniform)` is executed, Python runs the function code block with `n` equal to 100 and the name `generator_type` "bound" to the function `rng.uniform`.
 
-* While these lines are executed, the names `generator_type` and `np.random.uniform` are "synonyms", and can be used in identical ways.
+* While these lines are executed, the names `generator_type` and `rng.uniform` are "synonyms", and can be used in identical ways.
 
 This principle works more generally---for example, consider the following piece of code
 
@@ -507,7 +509,7 @@ factorial(4)
 
 The [binomial random variable](https://en.wikipedia.org/wiki/Binomial_distribution) $Y \sim Bin(n, p)$ represents the number of successes in $n$ binary trials, where each trial succeeds with probability $p$.
 
-Without any import besides `from numpy.random import uniform`, write a function
+Using `rng = np.random.default_rng()`, write a function
 `binomial_rv` such that `binomial_rv(n, p)` generates one draw of $Y$.
 
 ```{hint}
@@ -527,12 +529,12 @@ If $U$ is uniform on $(0, 1)$ and $p \in (0,1)$, then the expression `U < p` eva
 Here is one solution:
 
 ```{code-cell} python3
-from numpy.random import uniform
+rng = np.random.default_rng()
 
 def binomial_rv(n, p):
     count = 0
     for i in range(n):
-        U = uniform()
+        U = rng.uniform()
         if U < p:
             count = count + 1    # Or count += 1
     return count
@@ -558,7 +560,7 @@ Second, write another function that does the same task except that the second ru
 
 - If a head occurs `k` or more times within this sequence, pay one dollar.
 
-Use no import besides `from numpy.random import uniform`.
+Use `rng = np.random.default_rng()` to generate random numbers.
 
 ```{exercise-end}
 ```
@@ -573,15 +575,15 @@ Here's a function for the first random device.
 
 
 ```{code-cell} python3
-from numpy.random import uniform
+rng = np.random.default_rng()
 
 def draw(k):  # pays if k consecutive successes in a sequence
 
     payoff = 0
     count = 0
 
     for i in range(10):
-        U = uniform()
+        U = rng.uniform()
         count = count + 1 if U < 0.5 else 0
         print(count)    # print counts for clarity
         if count == k:
@@ -601,7 +603,7 @@ def draw_new(k):  # pays if k successes in a sequence
     count = 0
 
     for i in range(10):
-        U = uniform()
+        U = rng.uniform()
         count = count + ( 1 if U < 0.5 else 0 )
         print(count)
         if count == k: