Suppose that $X_1, ..., X_n$ are independent Bernoulli random variables with $\theta_i$ the success probability for $X_i$. Suppose these succes probabilities are related to a sequence of variables $t_1, ..., t_n,$ viewed as known constants, through $$\theta_i = \dfrac{1}{1 + \exp(-\alpha - \beta t_i)},\, i = 1,..., n.$$

Exercise: show that the joint densities of $X_1, ..., X_n$ form a two-parameter exponential family, and identify the statistics $\tau_1$ and $\tau_2$.

I know that:

A parametric family with parameter space $\Omega$ and density $f_X(x|\theta)$ w.r.t. a measure $\nu$ on $(\mathcal{X}, \mathcal{B})$ is called an exponential family if $$f_X(x|\theta) = c(\theta)h(x)\exp\bigg(\sum\limits_{i = 1}^k \pi_i(\theta)\tau_i(x)\bigg),$$ for some measurable functions $\pi_1, ..., \pi_n, \tau_1, ..., \tau_n$ and some integer $k$.

What I've tried to do: I think that in this case $f_X(x|\theta) = \prod\limits_{i =1}^n\theta_i^{x_i}(1-\theta_i)^{(1-x_i)}$. What follows is $$\prod\limits_{i =1}^n\theta_i^{x_i}(1-\theta_i)^{(1-x_i)} = \\ \theta_i^{\sum_{i = 1}^n x_i} (1-\theta_i)^{\sum_{i = 1}^n (1-x_i)} = \\ (\theta_i -\theta_i ^2)\exp\bigg(\sum_{i = 1}^n x_i + \sum_{i = 1}^n (1 - x_i)\bigg).$$

Question: Am I going in the right direction? What are $\tau_1 \text{ and } \tau_2$ in this case?

Thanks in advance!


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