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### 2406. `negative_binomial_distribution` should reject `p == 1`

**Section:** 26.6.9.3.4 [rand.dist.bern.negbin] **Status:** C++17
**Submitter:** Stephan T. Lavavej **Opened:** 2014-06-14 **Last modified:** 2017-07-30

**Priority: **3

**View all other** issues in [rand.dist.bern.negbin].

**View all issues with** C++17 status.

**Discussion:**

26.6.9.3.4 [rand.dist.bern.negbin] p2 requires "`0 < p <= 1`". Consider what happens when `p == 1`.
The discrete probability function specified by p1 involves "`* p^k * (1 - p)^i`". For `p == 1`, this is
"`* 1^k * 0^i`", so every integer `i >= 0` is produced with zero probability. (Let's avoid thinking about
`0^0`.)

Wikipedia states that `p` must be within
`(0, 1)`, exclusive on both sides.

**Previous resolution [SUPERSEDED]:**
Change 26.6.9.3.4 [rand.dist.bern.negbin] p2 as indicated: [*Drafting note*: This should be read as: Replace the
symbol "`≤`" by "`<`" — *end drafting note*]

explicit negative_binomial_distribution(IntType k = 1, double p = 0.5);

-2- *Requires*: `0 < p `~~≤~~< 1 and `0 < k`.

*[2014-11 Urbana]*

SG6 suggests better wording.

*[2014-11-08 Urbana]*

Moved to Ready with the node.

There remains concern that the constructors are permitting
values that may (or may not) be strictly outside the domain
of the function, but that is a concern that affects the
design of the random number facility as a whole, and should
be addressed by a paper reviewing and addressing the whole
clause, not picked up in the issues list one distribution
at a time. It is still not clear that such a paper would be
uncontroversial.

**Proposed resolution:**

This wording is relative to N4140.

Add a note after paragraph 1 before the synopsis in 26.6.9.3.4 [rand.dist.bern.negbin]:

A `negative_binomial_distribution` random number distribution produces random integers
$i\ge 0$
distributed according to the discrete probability function

$P\left(i\right|k,p)=\left(\genfrac{}{}{0ex}{}{k+i-1}{i}\right)\xb7{p}^{k}\xb7{(1-p)}^{i}$
.

[*Note*: This implies that $P\left(i\right|k,p)$ is undefined when `p == 1`. — *end note*]

*Drafting note*: $P\left(i\right|k,p)$ should be in math font, and `p == 1` should be in code font.