[ Berlin: Moved to NAD. N1932 makes this moot: the input_type template parameter has been eliminated. ]

Proposed resolution:

typedef  RealType  input_type;

Unlike all other distributions in TR1, this binomial_distribution has an implementation-defined input_type. We believe this is an unfortunate choice, because it hinders users from writing portable code. It also hinders the writing of compliance tests. We recommend instead:

typedef  RealType  input_type;

While this choice is somewhat arbitrary (as it was for some of the other distributions), we make this particular choice because (unlike all other distributions) otherwise this template would not publish its RealType argument and so users could not write generic code that accessed this second template parameter. In this respect, the choice is consistent with the other distributions in TR1.

We have two reasons for recommending that a real type be specified instead. One reason is based specifically on characteristics of binomial distribution implementations, while the other is based on mathematical characteristics of probability distribution functions in general.

Implementations of binomial distributions commonly use Stirling approximations for values in certain ranges. It is far more natural to use real values to represent these approximations than it would be to use integral values to do so. In other ranges, implementations reply on the Bernoulli distribution to obtain values. While TR1's bernoulli_distribution::input_type is specified as int, we believe this would be better specified as double.

This brings us to our main point: The notion of a random distribution rests on the notion of a cumulative distribution function, which in turn mathematically depends on a continuous dependent variable. Indeed, such a distribution function would be meaningless if it depended on discrete values such as integers - and this remains true even if the distribution function were to take discrete steps.

Although this note is specifically about binomial_distribution::input_type, we intend to recommend that all of the random distributions input_types be specified as a real type (either a RealType template parameter, or double, as appropriate).

Of the nine distributions in TR1, four already have this characteristic (uniform_real, exponential_distribution, normal_distribution, and gamma_distribution). We have already argued the case for the binomial the remaining four distributions.

In the case of uniform_int, we believe that the calculations to produce an integer result in a specified range from an integer in a different specified range is best done using real arithmetic. This is because it involves a product, one of whose terms is the ratio of the extents of the two ranges. Without real arithmetic, the results become less uniform: some numbers become more (or less) probable that they should be. This is, of course, undesireable behavior in a uniform distribution.

Finally, we believe that in the case of the bernoulli_distribution (briefly mentioned earlier), as well as the cases of the geometric_distribution and the poisson_distribution, it would be far more natural to have a real input_type. This is because the most natural computation involves the random number delivered and the distribution's parameter p (in the case of bernoulli_distribution, for example, the computation is a comparison against p), and p is already specified in each case as having some real type.