ArtsAutosBooksBusinessEducationEntertainmentFamilyFashionFoodGamesGenderHealthHolidaysHomeHubPagesPersonal FinancePetsPoliticsReligionSportsTechnologyTravel

Mild and Wild Randomness (normal distribution vs Cauchy distribution)

Updated on October 29, 2013

The first association on the term “randomness” is usually casino, toss of a coin, toss of a dice... One may think on casino as a place of huge uncertainty. However, this uncertainty is much more predictable and much less haphazard than some another random processes. Not all randomness are (qualitatively) equal!

In this article two types of randomness will be discussed through its representatives. For mild randomness we have bell shape distribution, called Gaussian distribution. On the opposite side, for wild randomness we have Cauchy distribution. In the fist case, there is negligible probability for extreme events whereas in the second case we have have so called “fat tail”. Although we can't know what will be the score of the next roulette game, neither we know the result of the next toss of a dice, in such a cases we have a macroscopic certainty.

So called 3-sigma rule. With the probability of 99% a normally distributed random variable will have a value within 3 standard deviation from the mean.
So called 3-sigma rule. With the probability of 99% a normally distributed random variable will have a value within 3 standard deviation from the mean.

Toss of a Coin and Blindfolded Archer's Score

What would be the number of score "6" after 600 toss of a dice? As we know, according to the definition, probability is the ratio of the number of desired outcomes and the number of possible events. Due to this fact we expect around 100 scores "6". What is our expectation if we toss a dice 6000 times? Again, we expect 1/6 of throws; and according to the low of large numbers, in this case, the number of desired outcomes will be closer (in percentage) to the figure of 1000 than in previous case to the figure of 100. So, the larger number of throws the smaller deviation from the expected value.

Let imagine a blindfolded archers trying to hit a target on a wall. For each snapshot, we measure the distance from the center of the target. What average distance one can expect after 100, 1000, 1000 shots? Is there any expected value? - in this case there is no any expected value. As the archer continue with snap shots, the average distance will change its value without tendency to any value. Namely, such a random process is essentially different than the process of toss of a coin. This is an example of a process where random variable has no expected value – as it is case with toss of a coin.

The first case is an example of so called mild randomness, meaning that there is a macroscopic certainty. We can't know the result of any particular experiment but we know the result of a large series of experiments. There is no possibility that one particular test significantly influence the majority of tests. However, in the second case there is no any certainty. There is always a possibility that the result of the next test radically influence the average of previous tests.

A distribution without the law of large numbers

In the experiment with a rotating line (see figure below), the random variable is the distance x. However, performing several series of measures, one can see that the average of x does not aspire to some constant value, i.e. the law of large numbers doesn't hold here.

History of Gaussian distribution

In the 19 th century one of a field of interest for mathematicians was calculation of orbital elements of celestial bodies (astronomy was a branch of mathematics in that time). It was known that telescopes are not absolutely precise but there is some systematic error in observation. This problem occupied both Gauss and Lagendre, and they developed (independently) a theoretical distribution that is nowadays called Gaussian distribution or normal distribution. Lagendre was first with his work (1805) while Gauss explain the theory better (1809).

It can be said that Gaussian distribution is widespread: it appears in several cases in mathematics, it has many nice mathematical properties (symmetry being the obvious one), it appears in quantum mechanics as well. However, there is no many random variables with normal distribution. In related books one can usually find the two examples: toss of a coin and toss of a dice. While it is possible to find normally distributed random variable in physics, it seems that such a distribution is not typical for socio-economical phenomena. For example, the speed of gas molecules is distributed as a Gaussian random variable. In his very popular book “The Black Swan”, N. Taleb said that randomness of Gaussian type are not present in socio-economical phenomena.

Gaussian distribution

The Gaussian distribution (or normal distribution) is a continuous probability distribution, defined by the probability density function (1) for standard case and (2) in general. Standard case means the average of 0 and standard deviation of 1. Notation of this distribution is N(μσ2).

The function is symmetric, having maximal value when x is the mean. The expected value of this distribution is the mean. Mean, median and mode for this distribution are all equal to each other. The inflection points of the curve are one standard deviation away from the mean, at points x=μ-σ, and x=μ+σ. It is known that with probability 95% a random variable distributed on this way will take a value within two standard deviation away from the mean. With probability of 99.7%, a random variable will takes the value in the interval of three standard deviation from the mean. Thus, after 3σ probability is negligible. This fact is known as "68-95-99 rule" or 3-sigma rule. (There is 68% probability that a random variable will takes the value within 1 sigma away from the mean.) Informally, this distribution is also called the bell curve.

The normal distribution.
The normal distribution.

Application of Gaussian distribution

This distribution appears theoretically in several parts of mathematics, including probability theory and Fourier analysis. It appears theoretically in quantum mechanics as well. In physics, velocity of molecules of gas is normally distributed. Based on this distribution, Albert Einstein explain the random motion of grains of pollen of the plant- which was firstly observed by Scottish botanist Robert Brown. The most known examples of a random processes that behave this way are toss of a coin and toss of a dice.

A short lesson on normal distribution

An example of Cauchy distribution: a line rotation around a point.
An example of Cauchy distribution: a line rotation around a point.

Cauchy distribution

In probability theory, the Cauchy distribution is a continuous probability distribution, defined by the probability density function (3) for standard case and (4) in general. This function is symmetric as well, having maximum at the median. The mean is not defined for this distribution while the mode is equal to the median.

Contrary to the Gaussian distribution within Cauchy distribution there is no expected value. Expected value for this distribution is not defined (since the definite integral defining expected value has no finite value). This means that in this case the low of large numbers doesn't hold. Because of this sensitivity on extreme events this distribution, as well as the others with the same property, is called the distribution with fat tail. This distribution is also known as Lorentz distribution of Cauchy-Lorentz distribution.

The Cauchy distribution.
The Cauchy distribution.
nr of series
1 mio
10 mio
100 mio
200 mio
An average distance x in an experiment with rotating line (see figure above).

Application of Cauchy distribution

One remarkable application of this distribution is in physics, where it describes amplitude in a resonance effect. Another one example of such a distribution is rotation of a line around a point (see figure), where the distance x is the random variable. If the angle is randomly chosen, then random variable x has the Cauchy distribution. The next table shows results of experiments showing there is no expected value. In the first column we see the number of series of tests whereas in every series we have 10 experiments. Obviously, there the low of large numbers doesn't hold in this case.

Besides, parameters of financial markets are examples of distribution with "fat tail" i.e. distribution with significan probability for extreme events.

An example: The Black Monday

The basic parameter on the financial markets is the price change. Usually we deal with a daily price change, that is expressed in percentage. Thus, the price change is a random variable, in a terminology of probability theory. The question is how this variable is distributed? Is it a kind of mild probability or wild?

It seems that price changes behave very unpredictable i.e. wild in this means. As we know, markets are very risk places with reported very extreme occurrences. Few times there were so huge changes on the market that that day are given names in a financial world. This is the case for October 19, 1987 that is called the black Monday. On this day markets throughout the world, from America and Europe to Asia, drooped for in most cased for 40%. New Zeland stock exchange reported even worse loss.

During the crisis in 1929 one of the worst day on markets was September 18, called the black Thursday.

A blindfold archer's snap shats are example of a so called wild randomness - a randomness where the law of large numbers doesn't hold.
A blindfold archer's snap shats are example of a so called wild randomness - a randomness where the law of large numbers doesn't hold.

A short quiz - check your knowledge!

view quiz statistics

Mild, slow and wild

Thus, mild randomness is every random process which can be described by normally distributed random variable. On the opposite side there is so called wild randomness, representing random processes with significant probability of extreme events. In case of Gaussian distribution, there is only 0.3% probability that random variable takes a value more than 3 standard deviation apart from the mean.

So, extreme event has probability which is in probability theory called practically impossible event. An example of wild randomness is Cauchy distribution, a distribution without defined the expected value and with property that events very far away from the median are possible.

In the prologue of his book "The misbehavior of markets", Benoit Mandelbrot says:

"Three states of matter – solid, liquid and gas – have long been known. An analogous distinction between three states of randomness – mild, slow and wild – arises from the mathematics of fractal geometry."

So, in addition to mild and wild randomness we deal with in this article, there is also something between, called "show" randomness. Actually, there is the whole class of distribution between Gaussian and Cauchy distribution, called L-stable distribution. More precisely, Gaussian and Cauchy distribution are on the marginal sides of these class.


    0 of 8192 characters used
    Post Comment

    • flysky profile imageAUTHOR


      6 years ago from Zagreb, Croatia

      Dear Larry, thank you very much for your comment and for your insight into this matter. Yes, as you said before statistics books deal only with Gaussian randomness. Nowadays, awareness that there are also essentially different distribution is growing. I would say that significant contribution in this understanding gave B. Mandelbrot – the father of fractal geometry, and N. Taleb – with his bestseller book "The Black Swan".

      I didn't know for this distribution, very interesting type and case of distribution! Thank you for sharing and enriching this subject with this impressive example.

    • Larry Fields profile image

      Larry Fields 

      6 years ago from Northern California

      Hi flysky. I don't know if this is still true. Many years ago, 'cookbook' statistics textbooks pretended that the distributions for most measurements were Gaussian. But as you point out, this distribution does not apply to many socioeconomic phenomena. And any social science study that makes the bureaucratic assumption of a Gaussian distribution is flawed. It would be more honest to use nonparametric statistics in the social sciences.

      My major professor (chemistry) gave an interesting example of a non-Gaussian distribution. A certain pipette manufacturer measures the volumes of water, which his pipettes draw. For the sake of argument, let's say that the distribution of pipette volumes is initially Gaussian.

      However this manufacturer throws out any pipette that is not within a specified range. The result is a bullet-shaped distribution of pipette volumes, which has no tails, and is definitely not Gaussian.

      Voted up and interesting.


    This website uses cookies

    As a user in the EEA, your approval is needed on a few things. To provide a better website experience, uses cookies (and other similar technologies) and may collect, process, and share personal data. Please choose which areas of our service you consent to our doing so.

    For more information on managing or withdrawing consents and how we handle data, visit our Privacy Policy at:

    Show Details
    HubPages Device IDThis is used to identify particular browsers or devices when the access the service, and is used for security reasons.
    LoginThis is necessary to sign in to the HubPages Service.
    Google RecaptchaThis is used to prevent bots and spam. (Privacy Policy)
    AkismetThis is used to detect comment spam. (Privacy Policy)
    HubPages Google AnalyticsThis is used to provide data on traffic to our website, all personally identifyable data is anonymized. (Privacy Policy)
    HubPages Traffic PixelThis is used to collect data on traffic to articles and other pages on our site. Unless you are signed in to a HubPages account, all personally identifiable information is anonymized.
    Amazon Web ServicesThis is a cloud services platform that we used to host our service. (Privacy Policy)
    CloudflareThis is a cloud CDN service that we use to efficiently deliver files required for our service to operate such as javascript, cascading style sheets, images, and videos. (Privacy Policy)
    Google Hosted LibrariesJavascript software libraries such as jQuery are loaded at endpoints on the or domains, for performance and efficiency reasons. (Privacy Policy)
    Google Custom SearchThis is feature allows you to search the site. (Privacy Policy)
    Google MapsSome articles have Google Maps embedded in them. (Privacy Policy)
    Google ChartsThis is used to display charts and graphs on articles and the author center. (Privacy Policy)
    Google AdSense Host APIThis service allows you to sign up for or associate a Google AdSense account with HubPages, so that you can earn money from ads on your articles. No data is shared unless you engage with this feature. (Privacy Policy)
    Google YouTubeSome articles have YouTube videos embedded in them. (Privacy Policy)
    VimeoSome articles have Vimeo videos embedded in them. (Privacy Policy)
    PaypalThis is used for a registered author who enrolls in the HubPages Earnings program and requests to be paid via PayPal. No data is shared with Paypal unless you engage with this feature. (Privacy Policy)
    Facebook LoginYou can use this to streamline signing up for, or signing in to your Hubpages account. No data is shared with Facebook unless you engage with this feature. (Privacy Policy)
    MavenThis supports the Maven widget and search functionality. (Privacy Policy)
    Google AdSenseThis is an ad network. (Privacy Policy)
    Google DoubleClickGoogle provides ad serving technology and runs an ad network. (Privacy Policy)
    Index ExchangeThis is an ad network. (Privacy Policy)
    SovrnThis is an ad network. (Privacy Policy)
    Facebook AdsThis is an ad network. (Privacy Policy)
    Amazon Unified Ad MarketplaceThis is an ad network. (Privacy Policy)
    AppNexusThis is an ad network. (Privacy Policy)
    OpenxThis is an ad network. (Privacy Policy)
    Rubicon ProjectThis is an ad network. (Privacy Policy)
    TripleLiftThis is an ad network. (Privacy Policy)
    Say MediaWe partner with Say Media to deliver ad campaigns on our sites. (Privacy Policy)
    Remarketing PixelsWe may use remarketing pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to advertise the HubPages Service to people that have visited our sites.
    Conversion Tracking PixelsWe may use conversion tracking pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to identify when an advertisement has successfully resulted in the desired action, such as signing up for the HubPages Service or publishing an article on the HubPages Service.
    Author Google AnalyticsThis is used to provide traffic data and reports to the authors of articles on the HubPages Service. (Privacy Policy)
    ComscoreComScore is a media measurement and analytics company providing marketing data and analytics to enterprises, media and advertising agencies, and publishers. Non-consent will result in ComScore only processing obfuscated personal data. (Privacy Policy)
    Amazon Tracking PixelSome articles display amazon products as part of the Amazon Affiliate program, this pixel provides traffic statistics for those products (Privacy Policy)