I am writing a document which often refers to two standard Probability Mass Functions (PMFs), one is a uniform distribution (e.g. U(1,5)=[0.2 0.2 0.2 0.2 0.2]) and the other has only a single non-zero element which is equal to 1 (e.g. [0 0 1 0 0]). I was wondering if there is a specific name or shorthand notation for this type of distribution?
The distribution is called "degenerate" (cf. http://en.wikipedia.org/wiki/Degenerate_distribution)
Do you mean a Bernouli distribution ie x is in {0,1} and with single parameter p where:
prob(X = 1) = p, prob(X = 0) = (1-p)
I'm thinking I might be misunderstanding here, as this seems to simple - are you possibly referring to a mixture of discrete uniform and bernouli (ie something comparable to zero inflated poisson...)
btw - I do think the name discrete uniform is probably a better description of the first distribution you describe as I believe the support for your random variables is {1,2,3,4,5} integers, since you put a point mass of .2 probability on each value, where as uniform usually refers to continuous valued random variables - ie
p(X=x) = 1/5 for 0
HI Milan, thanks for the reply. I think it may be even simpler than you think!
I omitted the word 'discrete' from my post as I thought this was implied by PMF (rather than PDF). A Bernouli distribution with p=1 is almost correct, however in my case x is not in the range {0, 1}, but {0...N}. So I have
prob(X=i)=1, prob(all other X)=0
For N=1 and i=1, this reduces to a Bernoulli distribution (with p=1). But for a discrete distribution of size N>1, does it have a common name?
If not then I will simply have to describe it myself, which is no problem, I just thought I'd seek the wisdom of others first :-)
Thanks again!
I think the issue here is that the distribution you are looking for is not a probability distribution. There is only one event possible, all others are impossible. There is no distribution. This is the extreme case where the concept of distribution collapses.
I would call it like "a single choice".
If there is uncertainty about which of the N events is the only possible, then you can again define a probability distribution - what will be Bernoulli with p(the selected event ist the possible) = 1/N and p(the selected event is one of the impossible ones) = (N-1)/N.
The distribution is called "degenerate" (cf. http://en.wikipedia.org/wiki/Degenerate_distribution)
Both distributions you specified are instances of the categorical distribution and they represent the two extremes. Think of it as an extension to the Bernoulli distribution for more than two categories. In the first case, your 5-sided dice is balanced and all faces are equiprobable.
If you are concerned with multiple draws from these distributions, your data will be drawn from the multinomial distribution, just like your data will be binomially distributed if each draw was from a Bernoulli distribution.
Specifically, I would refer to your probability mass function as the Kronecker delta function (http://en.wikipedia.org/wiki/Kronecker_delta).
A probability distribution concentrated in a point is called a Dirac measure. The Dirac measure concentrated in $x$ is denoted $\delta_x$ . Dirac's delta function and the Kronecker delta function are closely related concepts. The notion of the Dirac delta function is only used in a vector space. The Kronecker delta function can be used in a discrete setup, but formally it is a function and not a probability measure.
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