# Categorical Distribution

Suppose I want to do calculations with a random variable $X$ that has a simple categorical distribution (a.k.a generalised Bernoulli distribution or discrete distribution).

I was expecting to be able to work with a distribution "symbolically". That is, I was expecting that there would be a function like CategoricalDistribution, that we could use like this.

distX = CategoricalDistribution[{1/3, 1/2, 1/6}];


But no distributions in tutorial/DiscreteDistributions looked like they were suitable. An example of how the distribution would be used would be

{Probability[x == 0, x \[Distributed] dist],
Probability[x == 1, x \[Distributed] dist],
Probability[x == 2, x \[Distributed] dist]}


would output

{1/3, 1/2, 1/6}


In this particular case, we can do a workaround, by doing

dist = TransformedDistribution[
x1 + x2, {x1 \[Distributed] BernoulliDistribution[1/2],
x2 \[Distributed] BernoulliDistribution[1/3]}]


Unfortunately, this is not always possible, even with only 3 possible outcomes. For example, we cannot simulate CategoricalDistribution[{1/2, 0, 1/2}] like this, as we cannot have independent Bernoulli random variables that behave like this.

I was hoping the categorical distribution might be a special case of another distribution. Especially the MultinomialDistribution with 1 as its first argument looks useful, but I cannot transform that distribution into a distribution that gives an integer. Again it seems like I am missing a function like CategoricalDistribution to do so. To give an example

Position[RandomVariate@MultinomialDistribution[1, {1/2, 0, 1/2}],
1][[1, 1]]


"has the right distribution", but unfortunately I cannot use Position inside TransformedDistribution.

I am aware of functions like RandomChoice, but really I only want to do manipulations on distributions. Does anybody know how to do this?

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Strongly related – Jacob Akkerboom Mar 3 '14 at 14:38
I suppose dist = ProbabilityDistribution[Piecewise[{{1/2, y == 0}, {0, y == 1}, {1/2, y == 2}} ], {y, 0, 2, 1}], or dist = ProbabilityDistribution[Switch[y, 0, 1/2, 1, 0, 2, 1/2], {y, 0, 2, 1}]; do pretty much what I want. Maybe I should delete/close the question. – Jacob Akkerboom Mar 3 '14 at 14:46

EmpiricalDistribution can assign probabilities to each element in a set of discrete values:

Here's an example:

In[1]:= d = EmpiricalDistribution[{1/3, 1/2, 1/6} -> {1, 2, 3}];

In[2]:= Mean[d]
Out[2]= 11/6

In[3]:= PDF[d, x]
Out[3]= 1/3 Boole[1 == x] + 1/2 Boole[2 == x] + 1/6 Boole[3 == x]

In[4]:= CDF[d, x]
Out[4]= 1/3 Boole[1 <= x] + 1/2 Boole[2 <= x] + 1/6 Boole[3 <= x]

In[5]:= RandomVariate[d, 10]
Out[5]= {1, 3, 1, 2, 1, 2, 2, 1, 2, 1}

In[6]:= Probability[x == 1, x \[Distributed] d]
Out[6]= 1/3

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This is community wiki, i.e. people are welcome to edit and improve it. – Szabolcs Mar 3 '14 at 19:28
Wolfram should probably mention "categorical distributions" in the help page for EmpiricalDistribution, they are equivalent as far as I can tell. – A.G. Mar 3 '14 at 23:07
@A.G. Why don't you suggest it? There's a give feedback button at the bottom of the page. Maybe even link back here to prove that it's not easy to find. – Szabolcs Mar 3 '14 at 23:11
Done. Thanks for suggesting it. – A.G. Mar 5 '14 at 0:35