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Mean and Variance of Discrete Uniform Distributions

Posted on May 3, 2021 Written by The Cthaeh 1 Comment

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.

[Read more…]

Filed Under: Algebra, Probability Distributions Tagged With: Discrete uniform distribution, Expected value, Mean, Variance

Alternative Variance Formulas and Their Derivation

Posted on November 15, 2020 Written by The Cthaeh 2 Comments

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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Filed Under: Algebra, Measures, Probability Distributions Tagged With: Arithmetic, Expected value, Mean, Variance

Binomial Distribution Mean and Variance Formulas (Proof)

Posted on May 19, 2020 Written by The Cthaeh 31 Comments

A skyscraper resembling a binomial distribution

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.

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

[Read more…]

Filed Under: Algebra, Probability Distributions Tagged With: Expected value, Mean, Probability mass, Variance

The Binomial Distribution (and Theorem): Intuitive Understanding

Posted on May 19, 2020 Written by The Cthaeh 5 Comments

A skyscraper resembling a binomial distribution

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.

[Read more…]

Filed Under: Algebra, Combinatorics, Probability Distributions Tagged With: Bernoulli distribution, Binomial distribution, Coin flip, Mean, Probability mass, Variance

The Bernoulli Distribution: Intuitive Understanding

Posted on May 5, 2020 Written by The Cthaeh 1 Comment

A portrait of Jacob Bernoulli with a 17th century Swiss coin in the background

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.

[Read more…]

Filed Under: Probability Distributions Tagged With: Bernoulli distribution, Coin flip, Law of large numbers, Mean, Probability mass, Variance

Mean and Variance of Probability Distributions

Posted on August 28, 2019 Written by The Cthaeh 13 Comments

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.

[Read more…]

Filed Under: Probability Distributions Tagged With: Expected value, Law of large numbers, Mean, Probability density, Probability mass, Variance

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