In Multinomial Distribution, we have
\begin{align}
f(x_1,\ldots,x_k;n,p_1,\ldots,p_k) & {} = \Pr(X_1 = x_1\mbox{ and }\dots\mbox{ and }X_k = x_k) \\ \\
& {} = \begin{cases} { \displaystyle {n! \over x_1!\cdots x_k!}p_1^{x_1}\cdots p_k^{x_k}}, \quad &
\mbox{when } \sum_{i=1}^k x_i=n \\ \\
0 & \mbox{otherwise,} \end{cases}
\end{align}
where $x_i$ is an integer. Besides, we should note that x_i have a constant sum, and the sum of p_k equals to 1.0 (another constant sum).
But now, I need a Continuous Multinomial Distribution, where $x_i$ doesn't need to be an integer, and the sum of $x_i$ still equals $n$.
I cannot find such a distribution, could any one help me?
p.s. I found a related question in this site. Someone says that Dirichlet Distribution can be helpful. However, the alpha parameters in Dirichlet do not have a constant sum, which is not perfect for my problem.
Thanks very much!
Best Answer
PDF of the multinomial distribution can be evaluated outside its support, so we can define a distribution taking PDF as its analytic continuation. In fact, this new PDF
integrate to 1 on the corresponding stretched simplex for a binomial distribution.integrates to approximately 1 due to rectangular rule.