Conditional distribution of random sample from Poisson distribution

Question

I'm dealing with this problem for a while, and although I have some intuition, I think I'm missing something, please take a look:

We are given a random sample $X_1, X_2,...,X_n$ from Poisson$(\theta)$ distribution. I'm asked to prove that the conditional distribution of: $X_1, X_2,...,X_{n-1}$ given $Y=\sum_{i=1}^n X_i$ is multinomial.

What I have tried so far:

$P(X_1, X_2,...,X_{n-1} \mid Y = y) = \frac{P(X_1, X_2,...,X_{n-1},Y = y)}{P(Y=y)}$

Then, although I know the $n-1$ samples of $X_i$ are independent among each other, and that I could get their joint pdf easily, the problem comes up when I need to related such joint pdf to the pdf of $Y$. Since, I'm not sure whether they are independent, otherwise, the product of the pdfs would suffice.

I know that $Y \sim \text{Poisson}(n\theta)$.

Any clue? Thanks!

Answer 1

Hint

We have $$ P(X_1=x_1,\ldots, X_{n-1}=x_{n-1}, Y=y) = P(X_1=x_1,\ldots, X_{n-1}=x_{n-1}, X_n=y-\sum_{i=1}^{n-1}x_i) \\=f(x_1)f(x_2)\ldots f(x_{n-1}) f\left(y-\sum_{i=1}^{n-1}x_i \right)$$ where $$f(x) = \frac{\theta^xe^{-\theta}}{x!}$$