# Multinoulli Distribution Explanation

I know about multinoulli distribution, but I found a different explanation in a book that I have been reading and I didn't quite get it. It says:

The multinoulli, or categorical, distribution is a distribution over a single discrete variable with $$k$$ different states, where k is finite. The multinoulli distribution is parameterized by a vector $$p ∈[0,1]^𝑘-1$$, where $$p_i$$ gives the probability of the $$i$$-th state. The final, $$k$$-th state's probability is given by $$1−(1^𝑇)\cdot 𝐩$$. Note that we must constrain (1^𝑇)𝐩 ≤ 1.

I didn't understand how this represents the final state like this $$1−(1^𝑇)\cdot 𝐩$$ and anything about the vector.

If someone can provide detailed explanation it will be a real help.

• $1^T \mathbf{p}$ is just a "fancy" compact way to write $\sum_{i=1}^{k-1} \mathbf{p}_i$. (this is the dot product of the all-ones vector with the vector $\mathbf{p}$) – Clement C. Feb 11 at 6:47

here is a clarifying example using $$k = 3$$. Suppose that your experiment consists of drawing one ball. There are balls with three diverent colors: blue (b), green (g) and red (r). This corresponds to a multinoulli distribution with $$k = 3$$. We just have to specify the probabilities for the different outcomes. So we set $$p_b = \Bbb P (\text{Ball is blue}) = \frac 16 \quad \text{and} \quad p_g = \Bbb P (\text{Ball is green}) = \frac 13.$$
These two probabilities are enough to determine $$p_r = \Bbb P (\text{Ball is red})$$: $$p_r = 1 - (p_g + p_b) = \frac 12.$$ Or in more fancy notation if we set $$\mathbf{p} = (p_b, p_g)$$ and $$1^T = (1, 1)^T$$ then $$p_r = 1 - 1^T \cdot \mathbf{p}.$$