Assume that $Y_1,\dots, Y_n$ follows a Binomial distribution with probability of $p(d_i)$. Assume that the pdf of $Y_i$: $$ f(p_i,Y_i)=\binom{n_i}{y_i}p_i^{y_i}(1-p_i)^{n_i-y_i} $$
Assume that a model of $p(d)$ is $$ p(d)=\log\frac{p(d)}{1-p(d)}=\alpha+\beta d. $$
Question: I am confused about what is the Fisher information matrix?
I know the definition: the Fisher information of $\theta=(\alpha, \beta)$ is $$ I(\theta)=E\left[\left(\frac{\partial \log f(x;\theta)}{\partial \theta}\right)^2\right] $$
But I am not sure if I need to plug into all data sample $Y_1,\dots, Y_n$. I mean $$ I(\theta)=E\left[\left(\frac{\partial \log \prod_{i=1}^n f(x_i;\theta)}{\partial \theta}\right)^2\right] $$
My work:
Let $\theta=[\alpha, \beta]^\top$ and $z=[1,d]^\top$.
The score function is $$ S(\theta)=\sum_{i=1}^n\left[y_i-n_i \frac{e^{z_i^T\theta}}{1+e^{z_i^T\theta}}\right]z_i $$ and the Hessian matrix is $$ H(\theta)=-\sum n_ip_i(1-p_i)z_iz_i^\top. $$
