Problem 1.14 from Categorical Data Analysis 2nd. For the multinomial distribution, show that $$operatorname{corr}(n_j,n_k)=frac{-pi_jpi_k}{sqrt{pi_j(1-pi_j)pi_k(1-pi_k)}}$$ Show that $operatorname{corr}(n_1,n_2)=-1$ when $c=2$. The multinomial density is $$p(n_1,n_2,dots,n_{c-1})=binom{n!}{n_1!,dots,n_c!}pi_1^{n_1}dotspi_c^{n_c}$$ Let $n_j=sum_i y_{ij}$ where each $y_{ij}$ is Bernoulli with $E[y_{ij},y_{ik}]=0$, $E[y_{ij}]=pi_j$ and $E[y_{ik}]=pi_k$ Then $sum_j n_j=n$, with dimension $(c-1)$ since $n_c=n-(n_1+n_2+,dots,+n_{c-1})$. So each $n_jsim Bin(n,pi_j)$ $$begin{cases}E[n_j]=npi_j\ operatorname{Var}(n_j)=frac{pi_j(1-pi_j)}{n}end{cases}$$ then $$operatorname{corr}(n_j,n_k)=frac{-npi_jpi_k}{sqrt{npi_j(1-pi-pi_j)npi_k(1-pi_k)}}=frac{-pi_jpi_k}{sqrt{pi_j(1-pi_j)pi_k(1-pi_k)}}.$$ Is that … Read more