# Are all continuous random variables normally distributed?

All the time I see examples of the normal/Gaussian distribution with continuous random variables. So my question is do all continuous random variables have a Gaussian distribution?

• For some examples there a triangle, gamma and beta distributions and others? Aug 13, 2018 at 16:24
• @MichaelChernick Links help: triangle distribution, gamma distribution, beta distribution. :) Aug 13, 2018 at 16:28
• It's easy to construct a non-normally distributed continuous random variable: Draw any PDF-looking function with finite area underneath it, and normalize it so the area underneath it is 1. You can image a PDF that isn't bell-shaped. Aug 13, 2018 at 16:35
• Aug 13, 2018 at 19:13
• Imagine listening for the next peal of thunder in a thunderstorm. You'll usually wait only a few seconds, but might wait much longer. Clearly this cannot be symmetric, so cannot be normally distributed. Or consider the distribution of annual incomes -- in western countries, median income will be in the tens of thousands of dollars (a bit under \$60K in the US). But many incomes are well over ten times that average. Again, it won't be symmetric so cannot possibly be normally distributed. Aug 14, 2018 at 4:46

No.

Lots of real life variables have distributions which are better described as other distributions. t-distributions (heavier tails) are common, as are various skewed distributions, for example, many real measurements must be positive, so greater than or equal to zero, but can have a long tail of high values. Quite a lot of real world data is counts, or similar integer data, which is often better described by a Poisson distribution.

In my personal experience, in epidemiology, bio-medicine, and sociology, genuinely 'normal' distributions, that is real data which can best described as a normal distribution, are uncommon, but it does depend on the field you work in, and exactly what data you are looking at.

No.

There are many continuous probability distributions out of all the probability distributions. There are whole books containing nothing but such things.

Some of the non-normal continuous distributions introduced to new students of statistics include:

The normal/Gaussian distribution is important because of the Central Limit Theorem (CLT), which shows that for very many situations the sum of randomly distributed independent variables will tend to have a normal distribution, regardless of the constituent variables' original distributions. This can be useful for performing certain kinds of commonly-used statistical inference, which probably contributes to the frequency with which one encounters the normal/Gaussian distribution. Student's T distribution mentioned above gives some formalism to the "tend" in the CLT's "...will tend to have a normal distribution...", and is therefore also useful in these commonly used forms of statistical inference.

Not necessarily. The shape of the distribution is contingent on the continuous random variable's PDF - which isn't expressly Gaussian. Some counterexamples include the student-t distribution and the Laplace distribution.