# What distribution covers proportions of multiple outcomes?

I had a friend ask me this question yesterday and I'm not certain of the answer. She has data where the variable of interest is a proportion but there are three categories. So $y_{i1} + y_{i2} + y_{i3} = 1$. She's in a maximum likelihood estimation class and needs to know how to find the likelihood function for these data, but I didn't know the correct distribution.

EDIT: For example, election outcomes with three parties. So for each year (observation), party 1 gets $y_{i1}$% of the vote, party 2 gets $y_{i2}$% of the vote and party 3 gets $y_{i3}$% of the vote, with those three totaling to 100%.

• Could you provide an example on how does the data look like since it is not clear from your question? – Tim Dec 2 '15 at 13:57

First one is the categorical distribution. It describes an event that can take $k$ possible outcomes, $x \in \{1,...,k\}$, with probabilities $p_1,...,p_k$ such that $\sum_{i=1}^k p_i = 1$. You can think of it as of Bernoulli distribution with more than two categories. In this case your data is events and you are interested in probabilities.
On another hand, if you want to model the proportions of $k$ categories, than you could use Dirichlet distribution. It is a multivariate distribution for $k$ variables $p_1,...,p_k$ such that each $p_i \in (0, 1)$ and $\sum_{i=1}^k p_i = 1$. You can think of it as of a multivariate beta distribution. In this case your data are the proportions and you want to model their distribution.