One construct that’s extremely useful and provided by almost all programming languages is functions.
We have already met several functions, such as
sqrt()function from NumPy and
In this lecture we’ll treat functions systematically and begin to learn just how useful and important they are.
One of the things we will learn to do is build our own user-defined functions
We will use the following imports.
%matplotlib inline import numpy as np import matplotlib.pyplot as plt plt.rcParams['figure.figsize'] = (10,6)
A function is a named section of a program that implements a specific task.
Many functions exist already and we can use them off the shelf.
First we review these functions and then discuss how we can build our own.
4.2.1. Built-In Functions¶
Python has a number of built-in functions that are available without
We have already met some
Two more useful built-in functions are
bools = False, True, True all(bools) # True if all are True and False otherwise
any(bools) # False if all are False and True otherwise
The full list of Python built-ins is here.
4.2.2. Third Party Functions¶
If the built-in functions don’t cover what we need, we either need to import functions or create our own.
Examples of importing and using functions were given in the previous lecture
Here’s another one, which tests whether a given year is a leap year:
import calendar calendar.isleap(2020)
In many instances, it is useful to be able to define our own functions.
This will become clearer as you see more examples.
Let’s start by discussing how it’s done.
Here’s a very simple Python function, that implements the mathematical function \(f(x) = 2 x + 1\)
def f(x): return 2 * x + 1
Now that we’ve defined this function, let’s call it and check whether it does what we expect:
Here’s a longer function, that computes the absolute value of a given number.
(Such a function already exists as a built-in, but let’s write our own for the exercise.)
def new_abs_function(x): if x < 0: abs_value = -x else: abs_value = x return abs_value
Let’s review the syntax here.
defis a Python keyword used to start function definitions.
def new_abs_function(x):indicates that the function is called
new_abs_functionand that it has a single argument
The indented code is a code block called the function body.
returnkeyword indicates that
abs_valueis the object that should be returned to the calling code.
This whole function definition is read by the Python interpreter and stored in memory.
Let’s call it to check that it works:
4.4.1. Random Draws¶
Consider again this code from the previous lecture
ts_length = 100 ϵ_values =  # empty list for i in range(ts_length): e = np.random.randn() ϵ_values.append(e) plt.plot(ϵ_values) plt.show()
We will break this program into two parts:
A user-defined function that generates a list of random variables.
The main part of the program that
calls this function to get data
plots the data
This is accomplished in the next program
def generate_data(n): ϵ_values =  for i in range(n): e = np.random.randn() ϵ_values.append(e) return ϵ_values data = generate_data(100) plt.plot(data) plt.show()
When the interpreter gets to the expression
generate_data(100), it executes the function body with
n set equal to 100.
The net result is that the name
data is bound to the list
ϵ_values returned by the function.
4.4.2. Adding Conditions¶
generate_data() is rather limited.
Let’s make it slightly more useful by giving it the ability to return either standard normals or uniform random variables on \((0, 1)\) as required.
This is achieved in the next piece of code.
def generate_data(n, generator_type): ϵ_values =  for i in range(n): if generator_type == 'U': e = np.random.uniform(0, 1) else: e = np.random.randn() ϵ_values.append(e) return ϵ_values data = generate_data(100, 'U') plt.plot(data) plt.show()
Hopefully, the syntax of the if/else clause is self-explanatory, with indentation again delimiting the extent of the code blocks.
We are passing the argument
Uas a string, which is why we write it as
Notice that equality is tested with the
For example, the statement
a = 10assigns the name
ato the value
a == 10evaluates to either
False, depending on the value of
Now, there are several ways that we can simplify the code above.
For example, we can get rid of the conditionals all together by just passing the desired generator type as a function.
To understand this, consider the following version.
def generate_data(n, generator_type): ϵ_values =  for i in range(n): e = generator_type() ϵ_values.append(e) return ϵ_values data = generate_data(100, np.random.uniform) plt.plot(data) plt.show()
Now, when we call the function
generate_data(), we pass
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
While these lines are executed, the names
np.random.uniformare “synonyms”, and can be used in identical ways.
This principle works more generally—for example, consider the following piece of code
max(7, 2, 4) # max() is a built-in Python function
m = max m(7, 2, 4)
Here we created another name for the built-in function
max(), which could
then be used in identical ways.
In the context of our program, the ability to bind new names to functions means that there is no problem passing a function as an argument to another function—as we did above.
4.5.1. Exercise 1¶
Recall that \(n!\) is read as “\(n\) factorial” and defined as \(n! = n \times (n - 1) \times \cdots \times 2 \times 1\).
There are functions to compute this in various modules, but let’s write our own version as an exercise.
In particular, write a function
factorial such that
factorial(n) returns \(n!\)
for any positive integer \(n\).
4.5.2. Exercise 2¶
The binomial random variable \(Y \sim Bin(n, p)\) 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
binomial_rv such that
binomial_rv(n, p) generates one draw of \(Y\).
Hint: If \(U\) is uniform on \((0, 1)\) and \(p \in (0,1)\), then the expression
U < p evaluates to
True with probability \(p\).
4.5.3. Exercise 3¶
First, write a function that returns one realization of the following random device
Flip an unbiased coin 10 times.
If a head occurs
kor more times consecutively within this sequence at least once, pay one dollar.
If not, pay nothing.
Second, write another function that does the same task except that the second rule of the above random device becomes
If a head occurs
kor more times within this sequence, pay one dollar.
Use no import besides
from numpy.random import uniform.
4.6.1. Exercise 1¶
Here’s one solution.
def factorial(n): k = 1 for i in range(n): k = k * (i + 1) return k factorial(4)
4.6.2. Exercise 2¶
from numpy.random import uniform def binomial_rv(n, p): count = 0 for i in range(n): U = uniform() if U < p: count = count + 1 # Or count += 1 return count binomial_rv(10, 0.5)
4.6.3. Exercise 3¶
Here’s a function for the first random device.
from numpy.random import uniform def draw(k): # pays if k consecutive successes in a sequence payoff = 0 count = 0 for i in range(10): U = uniform() count = count + 1 if U < 0.5 else 0 print(count) # print counts for clarity if count == k: payoff = 1 return payoff draw(3)
0 1 2 3 0 0 1 0 1 2
Here’s another function for the second random device.
def draw_new(k): # pays if k successes in a sequence payoff = 0 count = 0 for i in range(10): U = uniform() count = count + ( 1 if U < 0.5 else 0 ) print(count) if count == k: payoff = 1 return payoff draw_new(3)
1 1 2 2 2 3 4 4 4 4