# What is the normal approximation of the multinomial distribution?

If there are multiple possible approximations, I'm looking for the most basic one.

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Let $P=(p_1,...,p_k)$ be the vector of your probabilities. Then the mean vector of the multivariate normal distribution is $np=(np_1,np_2,...,np_k)$. The covariance matrix is a $k \times k$ symmetric matrix. The diagonal elements are actually the variance of $X_i$'s; i.e.$np_i(1-p_i)$, $i=1,2...,k$. The off-diagonal element in the ith row and jth column is $\text{Cov}(X_i,X_j)=-np_ip_j$, where $i$ is not equal to $j$.