I have a question about the condition pmf of the multinomial distribution. I see some answers when the condition is given as equality for a certain variable, but could not see how it would be when it is given as an inequality. So, if each variable is given a condition less than a bound $k_n$,
First I define the multinomial distribution pmf as $$f(x_1,...,x_N) = \frac{m!}{x_1!\cdots x_N!}p_1^{x_1}\cdots p_N^{x_N}$$
Then, the conditional one with inequality conditions as $$f(X_1,...,X_N|X_1\leq k_1,...,X_N \leq k_N) = \frac{f((X_1=x_1,...,X_N=x_N) \cap (X_1\leq k_1,...,X_N \leq k_N))}{f(X_1\leq k_1,...,X_N \leq k_N)}$$
$$= \frac{f((X_1=x_1,...,X_N=x_N) \cap (X_1\leq k_1,...,X_N \leq k_N))}{ \sum\limits_{i=1}^{k_N} \sum\limits_{i=1}^{k_{N-1}}\cdots \sum\limits_{i=1}^{k_2}\sum\limits_{i=1}^{k_1} \frac{m!}{x_1!\cdots x_N!}p_1^{x_1}\cdots p_N^{x_N} }$$
I am not sure if the last equality is correct for the denominator and also what is the joint case in the numerator. It would be a great help if someone can give an answer or a help. Thank you!
The denominator is correct, and the numerator equals to either $f(x_1,\ldots,x_N)$ if $$x_1\leq k_1,\,\ldots,\,x_N\leq k_N$$ or to zero if at least one inequality is not fulfilled.