Question: (a) Suppose that \(X_{1}, X_{2}, \ldots, X_{n}\) are independent Poisson random variables where \(X_{i}\) has mean \(i \theta\), and \(\theta\) is an unknown parameter with parameter space \(\Theta=\{\theta: 0<\theta<\infty\}\).

1. Is this an exponential family? Justify your answer.
2. Is \(\sum_{i=1}^{n} X_{i}\) a sufficient statistic? Justify your answer.

(b) Suppose instead that \(X_{1}, X_{2}, \ldots, X_{n}\) are independent Bernoulli random variables where \(X_{i}\) has probability of success \(i \theta\), and \(\theta\) is an unknown parameter with parameter space \(\Theta=\left\{\theta: 0 \leq \theta \leq \frac{1}{n}\right\}\).

1. Is this an exponential family? Justify your answer.
2. Is \(\sum_{i=1}^{n} X_{i}\) a sufficient statistic? Justify your answer.

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