Goodness-of-Fit Test for Multinomial Distribution

Definition

Let $\mathbf{X}\sim \mathrm{Mult}(n,p_1,\dots ,p_{c-1})$ be a Multinomial Distribution and consider a null hypothesis $H_0:p_1=P_{10},p_2=P_{20},\dots ,p_c=P_{c0}$. Then, the test Statistic, which follows Chi-squared Distribution, is defined as $Q_{c-1}=\sum _{i=1}^c\frac{(X_i-n{p}_i{)}^2}{n{p}_i}\stackrel{D}{\to }{\chi }^2(c-1)$