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…]## 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.

## Intuitive Explanation of Expected Value

**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.