# Can a binomial distribution have negative x values?

The Poisson distribution makes a big deal that its great for modeling "Counts" of stuff (aka counts of monthly users) because its a non negative discrete distribution.

Well, aren't Binomial and Bernoulli distributions also non negative?

I can't think of any discrete distributions that can have negative x values. (Normal distribution i believe is a continuous distribution so it doesn't count)

• Counts are intrinsically non-negative. But other examples, such as the positions of a random walk with discrete steps certainly can. Commented Jan 9, 2023 at 1:19
• 1. 'The Poisson distribution makes a big deal that its great for modeling "Counts" of stuff (aka counts of monthly users) because its a non negative discrete distribution.'. No it doesn't, the Poisson is not able to take any such action. Some people might, but it would be quite odd for a person to single out the Poisson for this specific characteristic alone; which makes me wonder -- what are you reading?? Commented Jan 9, 2023 at 1:51
• 2. ""I cant think of any discrete distributions that can have negative x values." ... if you're asking for examples, one such (named) example is the Skellam (the difference of two independent Poisson r.v.s, with potentially different parameters). There are a number of questions on site that relate to it ... stats.stackexchange.com/search?q=skellam Commented Jan 9, 2023 at 1:54
• Note that the Poisson distribution is usually used not because it has non-negative support, but rather because the law of rare events (en.wikipedia.org/wiki/Poisson_distribution#law_of_rare_events) says that if we count an independent events each occuring with small probability, then the total count can be approximated with a Poisson distribution. Commented Jan 10, 2023 at 13:43

Poisson, binomial, Bernoulli, negative binomial, etc. are just model distributions - that is distributions that are analytically tractable and/or can be derived under rather simple assumptions. One could thus reformulate the question as:

Are there known model discrete distributions with a support containing negative numbers?

Then the question becomes obviously about how we define these discrete distributions - the most well-known ones are defined with support on non-negative integers. Specifically the binomial has support $$k\in\{0,1,...,n\}$$

As @Tim pointed out in their answer, we can easily define a distribution with negative support, using the one with positive support.

This brings us to the phenomena that we actually describe: thinking of a problem where negative counts arise naturally could suggest a distribution (which is not necessarily a known and/or model one.) However, many real counts are by nature positive - like the mentioned counts of monthly users (unless we explicitly shift the origin, just as in @Tim answer.)

Remarks:

• Good answer. So to summarize, in most real world situations, since discrete number counts itself is non-negative, the resulting distribution will also be non negative. Commented Jan 10, 2023 at 4:26
• @Katsu Correct: majority of practical situations concern counting items and events, whose number is by nature non-negative. There are situations when negative numbers are needed/useful - like price variation, temperature, etc., but they are rare, so no special model distributions were developed (or at least, they have not become famous.) Commented Jan 10, 2023 at 8:07

The binomial distribution is a distribution for a sum of Bernoulli trials: the sum of zeros and ones is non-negative. But it is not true that all discrete distributions are non-negative.

For a trivial example, say that $$X$$ follows a Poisson distribution and you create another variable $$Y = -X$$. $$Y$$ would be discrete and all its values would be smaller than or equal to zero by definition. The distribution for $$Y$$ does not have a name (it's like a Poisson distribution, just look at the $$-y$$ values), but as noted in the comments, there are named examples like the Skellam distribution.

• An even more trivial example is that opinion scores on an ordered scale could be taken as $-2(1)2$ or $-3(1)3$ with $0$ as a neutral category. It's just a matter of convention that people often or even usually work with $1(1)5$, $1(1)7$, or whatever. Commented Jan 9, 2023 at 10:19
• @NickCox Does $a(b)c$ mean integers from $a$ to $c$ inclusive in increments of $b$?
– J.G.
Commented Jan 10, 2023 at 15:55
• Precisely. It's an interesting question to me who introduced this notation (it is certainly in Tukey Annals of Mathematical Statistics 1948 as if self-evident) but it is standard in what I read and in software I use. Commented Jan 10, 2023 at 16:01

A less intuitive but important example for a discrete distribution with negative support would be the distribution of (unit or dollar) sales for a particular product at a given point of sale, conditional on various predictors like the day of the week, time of the year, promotions etc.

This is discrete because most products are sold by units (and therefore, dollar sales are also discrete). It has potential negative support because of product returns. Anecdotally, US consumer electronics retailers have quite a problem with big screen TVs that are bought right before the Super Bowl and returned for a full refund right after the game.

• Yeah... let me add to that "Anecdotally" comment, it's a real thing. Commented Jan 9, 2023 at 17:29
• @jbowman: thanks! Since the plural of "anecdote" is "data", two people with the same anecdote is a bona fide data point! :-) Commented Jan 10, 2023 at 11:34
• I've known people boasting about buying stuff with a card on one floor of a big store and getting a cash refund on another floor a few minutes later. Commented Jan 10, 2023 at 16:04