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Covariance and Correlation

Int this post, we introduce covariance and correlation, discuss how they can be used to measure the relationship between random variables, and learn how to calculate them. Covariance and Correlation are both measures that describe the relationship between two or more random variables. What is Covariance The formal definition of covariance describes it as

Introducing Variance and the Expected Value

In this post, we are going to look at how to calculate expected values and variances for both discrete and continuous random variables. We will introduce the theory and illustrate every concept with an example. Expected Value of a Discrete Random Variable The expected value of a random variable X is usually its mean.

The Law of Total Probability and Bayesian Inference

In this post we introduce the law of total probability and learn how to use Bayes’ rule to incorporate prior probabilities into our calculation for the probability of an event. Events don’t happen in a vacuum. They are related to and dependent on circumstances in their environment. Bayes rule allows us to relate the

Conditional Probability and the Independent Variable

In this post we learn how to calculate conditional probabilities for both discrete and continuous random variables. Furthermore, we discuss independent events. Conditional Probability is the probability that one event occurs given that another event has occurred. Closely related to conditional probability is the notion of independence. Events are independent if the probability of

Probability Mass Function and Probability Density Function

In this post, we learn how to use the probability mass function to model discrete random variables and the probability density function to model continuous random variables. Probability Mass Function: Example of a Discrete Random Variable A probability mass function (PMF) is a function that models the potential outcomes of a discrete random variable.

What is a Random Variable in Statistics: An Introduction to Probability

In this post, we look at the basic rules of probability and the notation used to describe probabilities and probability functions. We also introduce random variables. Probability is the branch of mathematics that deals with quantifying uncertainty. To model a real-world process that does not have a deterministic outcome, we can construct a probability

Calculus For Machine Learning and Data Science

This series of blog posts introduces multivariate calculus for machine learning. While the first few posts should be accessible to anyone with a high-school math background, the articles covering vector calculus require a basic understanding of linear algebra. If you need a refresher, I’d suggest you first check out my series on linear algebra.

The Fundamental Theorem of Calculus and Integration

In this post, we introduce and develop an intuitive understanding of integral calculus. We learn how the fundamental theorem of calculus relates integral calculus to differential calculus. Integration is the reverse operation of differentiation. Together they form a powerful toolset to describe non-linear functions. While differentiation enables us to describe the gradient or rate

Lagrange Multipliers: An Introduction to Constrained Optimization

In this post we explain constrained optimization using LaGrange multipliers and illustrate it with a simple example. Lagrange multipliers enable us to maximize or minimize a multivariable function given equality constraints. This is useful if we want to find the maximum along a line described by another function. The Lagrange Multiplier Method Let’s say

Understanding The Gradient Descent Algorithm

In this post, we introduce the intuition as well as the math behind gradient descent, one of the foundational algorithms in modern artificial intelligence. Motivation for Gradient Descent In many engineering applications, you want to find the optimum of a complex system. For example, in a production system, you want to find the ideal