Probabilistic World

Mean and Variance of Discrete Uniform Distributions

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A crowd of white Lego figures

In today’s (relatively) short post, I want to show you the formal proofs for the mean and variance of discrete uniform distributions. I already talked about this distribution in my introductory post for the series on discrete probability distributions. Well, this is a pretty simple type of distribution that doesn’t really need its own post, so I decided to make a post that specifically focuses on these proofs. More than anything, this is going to be a small exercise in algebra.

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

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Alternative Variance Formulas and Their Derivation

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The equations of three alternative variance formulas

In today’s post I want to show you two alternative variance formulas to the main formula you’re used to seeing (both on this website and in other introductory texts).

Not only do these alternative formulas come in handy for the derivation of certain proofs and identities involving variance, they also further enrich our intuitive understanding of variance as a measure of dispersion for a finite population or a probability distribution.

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The Birthday Problem: Analytic Solution

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UEFA Nations League semi-final between Netherlands and England (prematch, 06/06/2019), with the birthday problem formula in the background

Welcome to the first post from my new series. Here we’re going look at a famous probability question often called the birthday problem. This is actually a more general question related to the probability of at least one coincidence after a fixed number of draws from a discrete uniform distribution.

This post is part of my series Probability Questions from the Real World.

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Mean and Variance of Probability Distributions

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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.

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Intuitive Explanation of Expected Value

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A pile of poker chips and a few dice

Expected value is another central concept in probability theory. It is a measure of the “long-term average” of a random variable (random process). I know this doesn’t sound too clear, but in this post I’m going to explain exactly what it means.

There are many areas in which expected value is applied and it’s difficult to give a comprehensive list. It is used in a variety of calculations by natural scientists, data scientists, statisticians, investors, economists, financial institutions, and professional gamblers, to name just a few.

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