Generic operations common to all distributions are non-member functions

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##### Generic operations common to all distributions are non-member functions

Want to calculate the PDF (Probability Density Function) of a distribution? No problem, just use:

    pdf(my_dist, x);  // Returns PDF (density) at point x of distribution my_dist.

Or how about the CDF (Cumulative Distribution Function):

    cdf(my_dist, x);  // Returns CDF (integral from -infinity to point x)
                      // of distribution my_dist.

And quantiles are just the same:

    quantile(my_dist, p);  // Returns the value of the random variable x
                           // such that cdf(my_dist, x) == p.

If you're wondering why these aren't member functions, it's to make the library more easily extensible: if you want to add additional generic operations - let's say the n'th moment - then all you have to do is add the appropriate non-member functions, overloaded for each implemented distribution type.

| Tip
Random numbers that approximate Quantiles of Distributions If you want random numbers that are distributed in a specific way, for example in a uniform, normal or triangular, see Boost.Random. Whilst in principal there's nothing to prevent you from using the quantile function to convert a uniformly distributed random number to another distribution, in practice there are much more efficient algorithms available that are specific to random number generation.

For example, the binomial distribution has two parameters: n (the number of trials) and p (the probability of success on one trial).

The `binomial_distribution` constructor therefore has two parameters:

`binomial_distribution(RealType n, RealType p);`

For this distribution the random variate is k: the number of successes observed. The probability density/mass function (pdf) is therefore written as f(k; n, p).

| Note
Random Variates and Distribution Parameters Random variates and distribution parameters are conventionally distinguished (for example in Wikipedia and Wolfram MathWorld by placing a semi-colon (or sometimes vertical bar) after the random variate (whose value you 'choose'), to separate the variate from the parameter(s) that defines the shape of the distribution.

As noted above the non-member function `pdf` has one parameter for the distribution object, and a second for the random variate. So taking our binomial distribution example, we would write:

`pdf(binomial_distribution<RealType>(n, p), k);`

The ranges of random variate values that are permitted and are supported can be tested by using two functions `range` and `support`.

The distribution (effectively the random variate) is said to be 'supported' over a range that is "the smallest closed set whose complement has probability zero". MathWorld uses the word 'defined' for this range. Non-mathematicians might say it means the 'interesting' smallest range of random variate x that has the cdf going from zero to unity. Outside are uninteresting zones where the pdf is zero, and the cdf zero or unity.

For most distributions, with probability distribution functions one might describe as 'well-behaved', we have decided that it is most useful for the supported range to exclude random variate values like exact zero if the end point is discontinuous. For example, the Weibull (scale 1, shape 1) distribution smoothly heads for unity as the random variate x declines towards zero. But at x = zero, the value of the pdf is suddenly exactly zero, by definition. If you are plotting the PDF, or otherwise calculating, zero is not the most useful value for the lower limit of supported, as we discovered. So for this, and similar distributions, we have decided it is most numerically useful to use the closest value to zero, min_value, for the limit of the supported range. (The `range` remains from zero, so you will still get `pdf(weibull, 0) == 0`). (Exponential and gamma distributions have similarly discontinuous functions).

Mathematically, the functions may make sense with an (+ or -) infinite value, but except for a few special cases (in the Normal and Cauchy distributions) this implementation limits random variates to finite values from the `max` to `min` for the `RealType`. (See Handling of Floating-Point Infinity for rationale).

| Note

Discrete Probability Distributions

Note that the discrete distributions, including the binomial, negative binomial, Poisson & Bernoulli, are all mathematically defined as discrete functions: that is to say the functions `cdf` and `pdf` are only defined for integral values of the random variate.

However, because the method of calculation often uses continuous functions it is convenient to treat them as if they were continuous functions, and permit non-integral values of their parameters.

Users wanting to enforce a strict mathematical model may use `floor` or `ceil` functions on the random variate prior to calling the distribution function.

The quantile functions for these distributions are hard to specify in a manner that will satisfy everyone all of the time. The default behaviour is to return an integer result, that has been rounded outwards: that is to say, lower quantiles - where the probablity is less than 0.5 are rounded down, while upper quantiles - where the probability is greater than 0.5 - are rounded up. This behaviour ensures that if an X% quantile is requested, then at least the requested coverage will be present in the central region, and no more than the requested coverage will be present in the tails.

This behaviour can be changed so that the quantile functions are rounded differently, or return a real-valued result using Policies. It is strongly recommended that you read the tutorial Understanding Quantiles of Discrete Distributions before using the quantile function on a discrete distribtion. The reference docs describe how to change the rounding policy for these distributions.

For similar reasons continuous distributions with parameters like "degrees of freedom" that might appear to be integral, are treated as real values (and are promoted from integer to floating-point if necessary). In this case however, there are a small number of situations where non-integral degrees of freedom do have a genuine meaning.