The distribution is called a "categorical distribution", see

http://en.wikipedia.org/wiki/Categorical_distribution

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you can call this distribution a "categorical distribution"

http://en.wikipedia.org/wiki/Categorical_distribution

(better than "discrete distribution" as "categorical" implies that the number of outcomes is finite)

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Thank you. I hoped to have a specific name in contrast to the Bernoulli distribution, not as a generalization of the Bernoulli distribution.

One thing in the linked Wiki article is not clear to me: They write that there is a categorical distribution for k=1 (the domain is just {1}, comprising a single category with no altenative). I do not understand how this can be a probability distribution, since the i) there is nothing to "distribute over" and ii) the only allowed value is fixed, constant (=1). This "distribution" can not represent "knowledge" (nor uncertainty), The entropy is -1*log(1) = -1*0 = 0. This is all strange...

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to stress the difference with Bernoulli, you can use "generalized Bernoulli" ... and maybe "stricto sensu generalized Bernoulli" would be even more accurate !

when the name becomes longer than the proper mathematical definition, we have a problem ...

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the case k = 1 is a nice way to represent a deterministic event

however borderline it may seem, this k = 1 case not violate any axiom in the Kolmogorov formalism as far a s i can see

(now, it does not mean it is particularly useful or interesting ...)

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Thank you Fabrice!

An observation with K outcomes follows a Multinomial(1,pi1, pi2, ... pik) distribution. When K = 2, this reduces to the Bernoulli distribution.

Reference: Maydeu-Olivares, A. & Joe, H. (2006). Limited information goodness-of-fit testing in multidimensional contingency tables. Psychometrika, 71, 713-732.

p.716, available in ResearchGate

Alberto, for the multinominal distribution, the pi1, pi3, ...pik are independent and belong to different, independent nominal variables (possibly of the same kind, but this is not relevant). I was asking for a distribution referring to a single nominal variable with more than 2 levels. The resulting distribution is one-dimensional, and there is nothing like "contingency" that can apply here (because it is only about a single, one-dimensional variable).

I am sorry but the multinomial distribution is what are you looking for.

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bernoulli binomial

discrete distribution (or generalized bernoulli) multinomial

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Yes. Multinomial.  You can have multiway classification, but the underlying distribution is typically still multinomial,  If you have  a sample from the multinomial then you can think of this as bins with counts in the bins. This can then be called a I x 1  contingency table, where I is the number of levels. The word "level" can be confusing if there is no special order. With a two-way classification you get an I x J  table. etc. An important point is that you get a multinomial if the counts in the bins are independent Poisson random variables and you condition on the total count. 

From Wikipedia:

In probability theory, the multinomial distribution is a generalization of the binomial distribution. For n independent trials each of which leads to a success for exactly one of k categories, with each category having a given fixed success probability, the multinomial distribution gives the probability of any particular combination of numbers of successes for the various categories.

I possibly start to understand...

I see that the Bernoulli distirtbution is the "fundamental" distribution underlying the binomial distribution, so that the binomial reduces to the Bernoulli for n=1 "trial".

Similarily, the multinominal distribution can be considered for n=1 "trial". I was asking: has this special case (n=1) a special name (like in teh Bernoulli-binomial case), to express that we are not talking about a distribution that results from replicate trials (i.e., that it refers to one single trial).

Tosisng a coin is a Bernoulli trial, and the domain is {success, failure}, and the probability distribution is a Bernoully distribution Tossing a coin n times changes the domain to {0, 1, 2, ... n}, and the distribution is the binomial. Taking the Bernoully as the extreme case of the binomial means to translate the domain of the Bernoulli distribution to {0 ,1} (referring to the "number" of successes, what I think is not absolutely the same!).

Tossing a normal die with the six sides A-F is what trial? A multinomial? The domain is {A,B,...F}. Is such a distribuption really called multinominal? I don't think so. Even if one theoretically could call it like that, it would be uttery confusing, to my opinion (at least for me). Tossing a die n times provides a variable with the domain {(0,1,2,...n); (0,1,2...n); ... (0,1,2,...n)}. This is, to my knowledge, a multinominal distribution. For n=1 this would reduce to {(0,1);(0,1);...(0;1)} what I think is not exactly the same as {A,B,...F}, although it may be treated mathematically in the same way.

So although I see that the mathematical aspects can be formulated and treated just as a multinominal distribution with n=1, I see an important semantic difference, and I hoped that this difference is expressed be different names, like Bernoully vs. binomial.

Not sure if I made my point clear... I'd find it already difficult to explain such things in my own language, not to mention in a foreign language.

Jochen,

I am not sure what your question was, but as far as I read, you've been given some answers that agree the name you are looking for is "multinomial". I concur, and here are some complementary elements.

In the last post you say: "I see that the Bernoulli distribution is the "fundamental" distribution underlying the binomial distribution, so that the binomial reduces to the Bernoulli for n=1 "trial"." I think you've got it wrong on that one, the correct way to say it is the other way around:

I see that the Bernoulli distribution is the "fundamental" distribution underlying the binomial distribution, so that the *Bernoulli* is the *binomial* for n=1 "trial".

Indeed, if you add up Bernoulli variables you end up with a binomial variable. So, if you "add up" just one Bernoulli variable, it is theoretically speaking the simplest binomial variable.

I also think there is an error in your reasoning concerning the domain of the multinomial in the case of a die. You didn't make the error in the case of a Bernoulli variable, as you were saying the domain was {success, failure}. For a coin toss (an only, unique, coin toss) the domain is indeed {heads, tails} and the "number" of heads is a Bernoulli variable. If you toss the coin many (independent) times and that the probability of (say) heads is the same for all the tosses (i.e., no side of the coin is made heavier by a Nutella-dirty finger at some point during the coin tossing process), the random variable "number of heads in x tosses" is a binomial variable (i.e., it follows a binomial law) and its domain is {heads, tails} .

Likewise, if you throw a die the domain is {"1", "2","3", "4", "5", "6"}. Not "{(0,1,2,...n); (0,1,2...n); ... (0,1,2,...n)}." as you put it.

And if you toss the die just once, the domain is still {"1", "2","3", "4", "5", "6"}, and the distribution still multinomial (this is because the probability of getting a "1", or a "2" or a ... or a "6" is given by the probabilities that define your multinomial distribution.

If you think of a perfectly balanced (6-face) die for which the probability of each and every face is exactly of 1/6, the number of "1"s, of "2"s ... of "6"s follows a multinomial distribution the parameters of which are pi1=pi2=.... pi6= 1/6.

Not sure this answers your question (and sorry if I missed your point) but maybe it somehow helps.

Cheers,

Serban

Thank you Sebran, and every answer and contribution is helpful (be it as an bad example... what is not the case here. *kidding*).

So "multiominal" is the most comprehensive designation, comprising tossing a k-sided die n-times with k = (1,?) 2, 3, ... and n = 1, 2, ... (a "two-sided die" would be similar to a "coin"). From this point of view one can denote special cases. The special case of k=2 (and no restrictions on n) is called "binomial", the special case for k=2 and n=1 is named "Bernoulli".

My question was if the special case fo k>2 and n=1 has some own name. If I talk about a "multinominal" thing it typically means k>2 (otherwise usually the name "binomial" is used), but it also usually refers to n>1. If I specifically want to refer to a multinominal with n=1, is the a name addressing this directly or do I have to say something like "multiniminal for a single trial"?

I still find Fabrice's suggestion most appealing to call this a "categorical distribution".

My semantic problem I was trying to explaion is following:

The result of a Bernoulli trial is not a number. It is a dichotomous thing, something is there or not, yes or no, present or not, observerd or not. On can code this numerically (0/1), but the interpretation as a count for me is like interpreting some sign (some scribble, possibly a collection of random scratches on a surface) as a "word" (of a language; as a semaphore of information). Without even the possibility of othe words and a sender-reciever-system these are just scratches. Calling this a "word" in a language to what we have no knowledge seems to be an over-interpretation. If I see the result of a Bernoully trial as a count, then there is the expectation that such counts can be any integer number, at least theoretically. A restriction to an upper bound of 1 just removes all information of "counting" and leaves over only the information of a bit (binary variable). Interpreting this as a "count" seems to be an over-interpretation. Therefore, it makes a difference IMHO if the result is called a Bernoulli variable (that is about a bit) or a binomial variable (that is about a count).

Not sure if I could make this point clear. However, I have a similar "problem" for multinominal variables (specifically for those with k>2 ;) ), where I miss an appropriate nomenclature. 

Jochen

The answer for a k-sided die is multinomial. Is your question, if the die is one-sided, but the value can range from 1 to k, what is the distribution? If so, the answer is multinomial.

Could you give us some data that illustrates your question?

Joseph,

I have no data to illustrate the problem. Take the toss of a die. Or the draw of an item out of a box with k different items. Whatever.

I can always say the distribution is multinominal. That may be correct, but it does not help to distinguish the specific meanings of what one is talking about.

The outcome variable of a coin toss has a multinominal distribution. However, noone would say this. One could specify that is has a special multinominal distribution, which is the binomial distribution. And still this is confusing, because we are talking about a single toss. The most specific and clearest name is thus the Bernoulli distribution. The Bernoulli distribution is directly about a categorical random variable, about the categories itself. The random variable is a boolean variable.

The outcome variable of a series of coin tosses has a multinominal distribution, too. However, here the name "binomial distribution" is the most specific, the name makes a linguistic and well-understood distinction between a single toss as well as to a toss of a die with k>2 faces. In contrast to the Bernoulli distribution, the binomial distribution is about the counts and not about the categories, the random variable is an integer variable. 

The outcome variable of a single toss of a die has a multinominal distribution, too. Here I lack the linguistic possibility to dinstinguish between a single-toss experiment, where the actual face value is the random variable, and a series of tosses. where the number of occurences (of the different face values) is the random variable.

Jochen

Is your question

Is there a name for single-event probability with a possibility of k outcomes?

I do sense that is precisely his question. ;)

Yes :)

I suggest the name would be 'probability' or 'single-event probability'. The distribution is uniform unless the ki have additional constraints.

I believe one would have no problem to understand that Bernoulli multivariate would be a good title if the following notation is used.

Instead of one and zero for failure and success represent your sample space as follows

{(1;0),(0,1)}: (1,0) for failure and (0,1) to success.  Now consider the sample space as 

{(1.0,0).(0,1,0),(0,0,1)}.  Now consider u1, u2, u3, ...un a iid sample of such Bernoully of order 3 and dimension 2.  

Hence u1+u2+...+un =x is a trinomial with sample size n and eacn component is a frequency in the trinomial variable.

Note that in the first case of bernoulli, x would be a binomial (x1,x2) where x2=n-x1.

This notation is also good for the negative multinomial.

If the results of the randomized trial have the same opportunity (chance) to emerge, such as throwing the dice, the distribution of so-called the uniform distribution (discrete type).

I would call it the discrete uniform distribution.

Thank you for your comments. Just to make this point clear: I am not referring to a (discrete) uniform distribution. The shape of the distribution should not matter. A Bernoulli distribution can also be uniform, but the name "Bernoulli distribution" refers to all shapes of a binary distribution.

Further I think that "discrete distribution" does not hit the mark since it also includes the binomial and multinomial distributions, and we are left with the problem I tried to explain previousely.

For instance take the distribution of the nationality of a randomly selected person living today on this world. A uniform distribution is surely not very sensible, and based on Laplace's rule of indifference (of selecting the person), a reasonable probability distribution equals the distribution of the relative frequencies of the population sizes of the nations. When "selecting one particular person (at random)" is the experiment, the variable is the nationality of this person (categorical, can take the values {Afghanistan, Albania, ... Malawi, ... Zambia, Zimbabwe}). To my understanding, calling this a multinominal variable would imply that the result is a vector of numeric values (1 for the nation of the selected person and 0 otherwise). Although this is mathematically equivalent, is expresses a different point-of-view, and I am looking for a terminology that reflects this difference.

The distribution is called a "categorical distribution", see

http://en.wikipedia.org/wiki/Categorical_distribution

I think there might be some hare splitting going on. By convention, the phrase "random variable" refers to a numerical value. So, we need to identify categories with numbers in order to use the random variable terminology. Alternatively, you may use vectors. In your example the random vector (0, 1, 0, ...) is equivalent to Albanian, so why not just identify nationality with random vectors of 0s and 1s, where there is exactly one 1? The distribution of this is a special case of a multinomial with a sample size of 1. Likewise a Bernoulli variable is a special case of a binomial, and is also a special case of a multinomial if we change the notation to (0, 1) for a success and (1, 0) for a failure.

Terry, I am not so sure - why should "random variable" be neccesarily a numerical variable?

Place 1000 balls in a box. The balls have different colors and patterns. 500 balls are red with a white stripe. 80 are blue. 12 green with yellow dots. 187 black. 9 are purple, ... Does this distribution have a name?

Place 1000 balls in another box. Start with 8 red balls with a white stripe. 362 blue. ...

How would you describe each box?

Dear Joseph,

If each color have the same number of the balls, it will be distributed as a Uniformly distributed (discrete type), otherwise, it will be an arbitrary distribution (don't has name).

Joseph, your question seems to be similar to the following:

Take 1000 coins (or red and white balls) lying box 1. 372 show "head" (or are red), the others "tail" (or are white). Does this distribution habe a name? - Yes! It is called a "Bernoulli distribution".

Take 1000 coins lying in box 2. 711 show "head". This is again a Bernoulli distribution.

The only difference is the the parameter value is different.

In your example, a box contains more than just two different kinds of things. The distribution (whatever its name is, presumably "categorical distribution") has k-1 parameters, where k is the number of different itmes (categories). For the case of the Bernoulli distribution, k=2 and the distribution has k-1 = 1 parameter. Thus, having a "categorical distribution" with many categories is simply is cumbersome to decribe the particular instance* of the distribution (unless it is uniform or has some other easy-to-explain pattern).

The same applies to any other distribution. Take the normal distribution. The name of this distribution is "normal distribution" or "Gauss distribution" (some other names were sometimes proposed like e.g. "central distribution"). This is the name. The distribution has two parameters, so the communication of a particular instance of this distribution is quite simple.

*I am not sure if "instance" in a correct term here but I hope is is clear what I mean.

Huda, I think "uniform (discrete)" describes the fact but is not a name but rather a... well... description. It describes two properties: the shape and the kind of support. The Beta distribution for instance nac also have all kinds of shapes, also uniform B(1,1). Hoever, the name of this distribution is just "beta distribution" and not "uniform (continuous with support in 0

Jochen

How does the box distribution differ from selecting one person at random?

I am not sure what you are aiming at ... I don't see a difference. The items in the box have a frequency distribution, and this can be translated 1:1 into a probability distribution. In either case one can talk about a distribution. When taking about a single person (or item), there is no frequency distribution, there is "only" a probability distribution. This should not matter for discussing the name of a distribution (be it a frequency or probability distribution). Or do I miss the point completely?

There is no difference. I was not sure you agreed there was no difference. It is possible to know the distribution of people with an accurate census (not easy, but possible). So people are no different from the box example. We could know the frequency distribution of people.

My point is that if you have a name for the box distribution you have an answer to, "Does this probability distribution have a name?"

Ok, I see. But the name I am looking for does not refer to a box having a particular set of items. It should correctly label all the boxes containg k>2 different kinds of items.So far I think that "categorical distribution", or better "k-categorical distribution" (where k gives the number of categories) is a good name.

Yes. k-categorical distribution sounds right.

My example was of two boxes with the intended extension to n boxes. That makes us in agreement. 

There is really no good reason that a random variable has to be numeric except that it's standard terminology. However, since any categorical variable can be mapped onto numbers or vectors, it doesn't really matter.

My main point was that we can get by with existing names. Maybe categorical variable would be best.

A multinomial distribution.

(warning: do not feed the troll!)