Moment Estimator in Multivariate Hypergeometric Distribution

Question

We suppose that a population contains $K$ different types of object, with R.V $X_1$, $X_2$, ... , $X_K$, being the number of each type, and we have $\sum_{k = 1}^{K}X_{k} = N$. Then, a simple random sample of size $n$ is taken and the number of each type, $Y_1$, $Y_2$, ... , $Y_K$ is obtained so that $\sum_{k = 1}^{K}Y_{k} = n$. I noticed that this can be modeled by a multivariate hypergeometric distribution: $$ P(Y_1, Y_2, ... ,Y_K|x_1, x_2, ... , x_K) = \frac{\prod_{k = 1}^{K} {x_k \choose y_k}}{{N \choose n}}, $$ with mean and variances $$ E(Y_k|x_k) = n\frac{x_k}{N} \quad \text{and} \quad Var(Y_k|x_k) = n\frac{x_k}{N}(1-\frac{x_k}{N})\frac{N-n}{N-1}. $$ Assuming that the numbers of $X_1, X_2, ... , X_K$ are unknown but the $N$ is known. I am wondering how can I obtain the method of moments estimator for $X_k$ and the variance of this estimator along with an estimator of this variance?