This is a bonus post for my main post on the binomial distribution. Here I want to give a formal proof for the binomial distribution mean and variance formulas I previously showed you.

[Read more…]## The Binomial Distribution (and Theorem): Intuitive Understanding

Hi, everyone! And welcome to my post about the binomial distribution! Just like the Bernoulli distribution, this is one of the most commonly used and important discrete probability distributions.

*This post is part of my series on discrete probability distributions.*

## The Bernoulli Distribution: Intuitive Understanding

In today’s post, I’m going to give you intuition about the Bernoulli distribution. This is one of the simplest and yet most famous discrete probability distributions. Not only that, it is the basis of many other more complex distributions.

*This post is part of my series on discrete probability distributions.*

## Discrete Probability Distributions: Overview (Series)

In my previous two posts I sketched the frame of the big picture around **probability distributions**. In my introductory post I gave some intuition about the general concept and talked about the two major kinds: **discrete** and **continuous** distributions. And in the follow-up post I related the concepts of mean and variance to probability distributions. I showed that this connection itself goes through two fundamental concepts from probability theory: the law of large numbers and expected value.

Now I want to build on all these posts. My plan is to start introducing commonly used discrete and continuous distributions in separate posts dedicated to each one. And I want to start with the former, since they are significantly easier to understand.

The goal of the current post is to be a final warm-up before delving into the details of specific distributions.

[Read more…]## Mean and Variance of Probability Distributions

In my previous post I introduced you to **probability distributions**.

In short, a probability distribution is simply taking the whole probability mass of a random variable and distributing it across its possible outcomes. Since every random variable has a total probability mass equal to 1, this just means splitting the number 1 into parts and assigning each part to some element of the variable’s sample space (informally speaking).

In this post I want to dig a little deeper into probability distributions and explore some of their properties. Namely, I want to talk about the measures of central tendency (**the** **mean**) and dispersion (**the** **variance**) of a probability distribution.

## Introduction to Probability Distributions

If you want to take your understanding of probabilities to the next level, it’s crucial to be familiar with the concept of a **probability distribution**.

In short, a probability distribution is an __assignment of probabilities or probability densities__ to __all possible outcomes__ of a __random variable__.

For example, take the random process of flipping a regular coin. The outcome of each flip is a random variable with a probability distribution:

- P(“Heads”) = 0.5
- P(“Tails”) = 0.5

Depending on the type of random variable you’re working with, there are two general types of probability distributions: **discrete** and **continuous**. In this post, I’m going to give an overview of both kinds. And in follow-up posts I’m going to individually introduce specific frequently used probability distributions from each kind.