Is there a Continuous Multinomial Distribution?? 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!
 A: 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.
A: Here's a distribution which seems to satisfy your criteria, but which I very much suspect won't be what you're looking for.  However, refining your criteria to rule it out might help clarify what it is that you are looking for.  Let $\ A\ $ be a finite subset of $\ \left\{\alpha\in\mathbb{R}^k\,\vert\ \sum_j\alpha_j=n , \alpha_i\not\in\mathbb{Z}, \alpha_i > 0 \mbox{ for all } i\right\} $ and $\ p_i, i=1,2,\dots,k,\ $ be real numbers with $\ 0<p_i\le 1, \sum_i p_i=1\ $, and $$\ G\ \  =\ \sum_{\alpha\in A}\frac{\Gamma(n+1)}{\Gamma(\alpha_1+1)\Gamma(\alpha_2+1)\dots\Gamma(\alpha_k+1)}\,p_1^{\alpha_1} p_2^{\alpha_2}\dots p_k^{
\alpha_k}\ .$$
Then take
 \begin{align}
f(&\alpha_1,\ldots,\alpha_k;n,p_1,\ldots,p_k) {} = \Pr(X_1 = \alpha_1\land\dots\land X_k = \alpha_k) \\  \\
& {} = \begin{cases} G^{-1}\frac{\Gamma(n+1)}{\Gamma(\alpha_1+1)\Gamma(\alpha_2+1)\dots\Gamma(\alpha_k+1)}p_1^{\alpha_1} p_2^{\alpha_2}\dots p_k^{
\alpha_k}, &
\mbox{when } \alpha\in A \\  \\
0 & \mbox{otherwise,} \end{cases}
\end{align}
A: I am not aware of the distribution you were looking for, but there is a "continuous" categorical distribution (special case of multinomial distribution with $n=1$) recently popped up in the machine learning community.
