When we look at data, we are often not interested in every individual fluctuation. We want to see the bigger picture: is this growing? Is this declining? Where is it heading? The moving average is one of the simplest tools we have to answer that question. Before we define it formally, let us build the intuition from the ground up.

Consider those three data points: 10, 20, 10

20 is like a bump or big increase in our data. Now average those three values.

Real-World Analogies

Think about how you pay for services in everyday life.

A post-paid phone plan is a perfect analogy for an annuity immediate. You make calls throughout the month, and only at the end of the period does the carrier tally up your usage and send you a bill. You consume first, pay later.

Conversely, a prepaid internet plan mirrors an annuity due. You load credit before the service begins. If you want internet access for the month, you pay upfront

This article introduces the accumulation function and effective interest rates through concrete examples, showing how we measure investment growth over time. Breaking down total growth into period-by-period factors allows us to better understand how an investment performs and compare growth rates across different time periods.

Starting with Numbers

Let me define a simple amount function at different time periods:

$$A(0) = 100, \quad A(1) = 110, \quad A(2) = 121, \quad A(3) = 133.1, \quad A(4) = 146.41$$

What’s the growth factor between $t=0$ and $t=1$? What number do you need to multiply $A(0)$ by to get to $A(1)$?

As a health-conscious person, you’re interested in tracking your weight. You decide to step on the scale every day and record the result. After three days, your data might look like this:

$$w(1) = 150, \quad w(2) = 151, \quad w(3) = 149$$

where the numbers in parentheses represent the day number, and the values represent your weight in pounds.

This creates a weight function, denoted $w(d)$. This function takes a day number $d$ as input and returns your weight on that particular day as output. In other words, $w(d)$ tells you your total weight on day $d$.

When learning hypothesis testing, have you ever wondered why sometimes we use one-tail tests instead of two-tail tests? One of the biggest sources of confusion is figuring out which one to use. Let me show you what goes wrong when you pick the wrong test.

Let’s say we have two classes of students and we compute their mean GPA. Our question is: Is there any significant difference in mean GPA between class A and class B? Notice the key word here is “difference.” We want to know if they’re different, not specifically if one is higher than the other. We’ll test this at the 5 percent significance level.

Consider the function $f: \mathbb{N} \to \mathbb{R}$ where

$$ f(x) = x^2 $$

This is a simple quadratic equation you’re acquainted with, but note that our domain are only natural numbers, meaning $1, 2, 3, 4, \ldots$. The graph of this function looks like this:

But this may not be the graph of the most common version of this quadratic. Indeed, it’s more commonly defined as a function from $f: \mathbb{R} \to \mathbb{R}$. While the natural numbers are said to be countable, the real numbers are uncountable.

Understanding Limits: Targets and Approaches

Introduction

When you first learned about limits in calculus, you likely encountered two different-looking definitions that your textbook claimed were “equivalent.” One used sequences and indices, while the other used those mysterious $\delta$ and $\epsilon$ symbols. Perhaps you wondered: Why do we need two definitions for the same concept? Are they really the same?

This article explores the conceptual machinery behind limits by introducing two intuitive ideas: targets (where we want our function values to land) and approaches (how we get close to a point in the domain). We’ll see that these definitions are equivalent in $\mathbb{R}$, but for a subtle and beautiful reason that connects to the Archimedean principle and the countable structure of the natural numbers.

The outcomes of a sample space (see the article on sample spaces referenced earlier) are not necessarily numerical. However, as mathematicians, we are interested in quantities such as averages, variances, and distributions, all of which require numbers.

This is where random variables come in.

A random variable is a function that maps outcomes of a sample space to real numbers. If the sample space is denoted by $\Omega$, then a random variable is a mapping

Let $\{a_n\}_{n=m}^{\infty}$ be a sequence of reals. Think of $m$ as the starting index of the sequence. Here are some sequences with different starting indices:

For $m=1$:

$$ a_1, a_2, a_3, a_4, a_5, \ldots $$

For $m=3$:

$$ a_3, a_4, a_5, a_6, a_7, a_8, \ldots $$

Now consider a natural number $N \geq m$. A tail of our sequence $\{a_n\}_{n=m}^{\infty}$ is defined as:

$$\{a_n\}_{n=N}^{\infty}$$

Visual Illustration

Let’s take $m=1$ and consider the original sequence starting from index 1:

If you interrogate the formula for the dot product, it will gift to you all the explanations you need. You’ve seen mathematicians define the same dot product in different ways.

The Geometric Definition of the Dot Product

In the two-dimensional plane, start with two vectors at the origin. Think of our vectors as line segments that we draw starting at coordinate $(0,0)$ (your origin). The dot product of those two line segments (vectors) equals: