Could the Real “Probability Density Function” Please Stand Up

This is a quicky post, not even worthy to be a full post in and of itself because the topic is so short. However, I’m hoping that the wording and content might lead a weary Data Analyst traveler who is just trying to find the correct Probability Density Function, when the web insists on showing multiple formulae, to the answer they seek.

For Gaussian/Normal Distributions I found 3 different formulae. Except, because of algebra, I didn’t. They are:

1: f(x \space | \space \mu, \sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}​} [1]

2: f(X) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(X-\mu)^2}{2\sigma^2}​} [2]

3: f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2​} [3]

I wrestled over these for a short spell, until I realised that they are all the same, just expressed slightly differently. If you use the rules of algebra appropriately you can change any one of them into any of the others. Personally, I think the first one is the tidiest looking, so I’m going to opt for that, but they’re all the same, and should not be cause for concern. I hope this helps allay at least one persons concerns…


  1. This was included in the Udacity “AWS Machine Learning Foundations Course” content. I can’t link directly to it as a result.
  2. Ireland’s State Examination Commission’s booklet of Formulae and Tables
  3. Wikipedia article on Normal Distribution

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