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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?

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    $\begingroup$ For some examples there a triangle, gamma and beta distributions and others? $\endgroup$ Aug 13, 2018 at 16:24
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    $\begingroup$ @MichaelChernick Links help: triangle distribution, gamma distribution, beta distribution. :) $\endgroup$
    – Alexis
    Aug 13, 2018 at 16:28
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    $\begingroup$ 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. $\endgroup$ Aug 13, 2018 at 16:35
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    $\begingroup$ 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. $\endgroup$
    – Glen_b
    Aug 14, 2018 at 4:46

3 Answers 3

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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.

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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.

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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.

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