Can a binomial distribution have negative x values?

Question

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. — Henry, Jan 09 '23 at 01:19

Roger V. · Accepted Answer · 2023-01-10T10:46:10.477

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:

Wikipedia has a list of discrete probability distributions, many of which are the distributions on integers. On a brief glance none of them has support extending to negative numbers.
A similar question: Can the discrete variable be a negative number?

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. — Katsu, Jan 10 '23 at 04:26
@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.) — Roger V., Jan 10 '23 at 08:07

score 9 · Answer 2 · edited Jan 09 '23 at 10:17

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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.

edited Jan 09 '23 at 10:17

Nick Cox

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answered Jan 09 '23 at 09:17

Tim

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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. – Nick Cox Jan 09 '23 at 10:19
@NickCox Does $a(b)c$ mean integers from $a$ to $c$ inclusive in increments of $b$? – J.G. Jan 10 '23 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. – Nick Cox Jan 10 '23 at 16:01

score 3 · Answer 3 · answered Jan 09 '23 at 15:58

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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.

answered Jan 09 '23 at 15:58

Stephan Kolassa

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Yeah... let me add to that "Anecdotally" comment, it's a real thing. – jbowman Jan 09 '23 at 17:29
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@jbowman: thanks! Since the plural of "anecdote" is "data", two people with the same anecdote is a bona fide data point! :-) – Stephan Kolassa Jan 10 '23 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. – Nick Cox Jan 10 '23 at 16:04

Can a binomial distribution have negative x values?

3 Answers3