# Indicator function for Probability Distributions

I have come across a couple of formula for definition of probability distributions that use an indicator function, e.g. $\mathbb{I}(X=x)$:

A couple of examples:

1)

\begin{align} \mathbf{N}_t\ &=\ \left( \sum_{i=1}^{N} \mathbb{I}(X_{it}=1), \sum_{i=1}^{N} \mathbb{I}(X_{it}=2), \sum_{i=1}^{N} \mathbb{I}(X_{it}=3), \sum_{i=1}^{N} \mathbb{I}(X_{it}=4) \right) \\[.3cm] \hat{\boldsymbol{\theta}}_t\ &=\ \frac{\mathbf{N}_t}{N} = \left( \frac{N_{1t}}{N}, \frac{N_{2t}}{N}, \frac{N_{3t}}{N}, \frac{N_{4t}}{N} \right) \end{align}\quad,

2)

$$\mathrm{Mu}(\mathbf{x}|1,\boldsymbol{\theta})\ =\ \prod_{j=1}^{K} \theta_{j}^{\mathbb{I}(x_j=1)}\quad,$$

and 3)

\begin{align} \mathrm{Dir} (\mathbf{x}|\boldsymbol{\alpha})\ &= \ \frac{1}{B(\boldsymbol{\alpha})} \prod_{k=1}^{K} x_k^{\alpha_k-1} \mathbb{I}(\mathbf{x} \in S_K)\quad,\\ &\qquad \mathrm{where}\ \ S_K = \{ \mathbf{x} : x_k \le 1,\; \sum_{k=1}^Kx_k=1 \} \end{align}

I understand this is a set function, which is $1$ or $0$ (probably) depending on a condition. In equation (1), I have no problem to understand $\mathbf{N}_t$. In (2) and (3), I am not that sure. I guess in (2) for a special case of multinomial distribution, that is multinoulli, indicator function returns only one parameter from a vector. Is this form also equivalent to this one:

$$\mathrm{Mu}(\mathbf{x}|1,\boldsymbol{\theta}) \ = \ \prod_{j=1}^{K}\theta_{j}^{x_j}\quad \mathrm{?}$$

I don't understand the meaning of indicator function in the the third example. Could someone help me with that?

Thanks

In the third example, the indicator function is 1, if $0 \leq x_k \leq 1, k=1,...,K$ and $\sum_{k=1}^K x_k = 1$, otherwise it is zero.

In particular: When sampling from a Dirichlet distribution, you will almost surely get a probability vector.