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A random sample of $6$ observations $(X_1, X_2, \cdots, X_6)$ is generated from a Geometric($\theta$), where $\theta \in (0, 1)$ unknown, but only $T = \sum_{i=1}^{6} X_i$ is observed by the statistician.

(a) Describe the statistical model for the observed data ($T$)

(b) (i) Is it possible to parameterize the model by $\Psi = \frac{1-\theta}{\theta}$ ? Prove your answer

(ii) Is it possible to parameterize the model by $\Psi = \theta(1-\theta)$ ? Prove your answer

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closed as unclear what you're asking by Brian Borchers, Cesareo, Lee David Chung Lin, metamorphy, Gibbs Jan 23 at 21:04

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Watch out! The fact that $X_i \sim Geo(\theta)$ by no means implies that the sum of i.i.d. geometric random variables (i.e. $T$) has a geometric distribution!

Recall that the terminology random sample implies that the observations have the distribution that you have mentioned and are also conditionally independent.

Recall also the definition of statistical model for random samples.

Considering the random sample $\mathbf{X}_n=\{X_1,\dots,X_n\}$, let $\mathcal{X}$ be the sample space for one of the random variables and $\Theta$ the parameter space. Any statistical model for random samples has the following form

$$\left\{\mathcal{X^n} \quad; \quad f_n(\mathbf{x}_n;\theta) = \prod_{i=1}^n f_{X}(x_i;\theta) \quad ; \quad \theta\in\Theta\right\}$$

where $f_X$ is the probability law of one of the random variables in the random sample and $f_n$ is the joint probability of the random vector $\mathbf{X}_n$. That is, a triplet consisting of a sample space (for the random vector $\mathbf{X}_n$), a joint parametric probability distribution and a parameter space on which the family of distributions is parametrized.

So, steps to solve your problem:

1) Identify the correct sample space for your random variable $T$.

2) Determine the joint distribution of $\mathbf{X}_n$ by considering the independence and identical distribution of the random variables $X_1, X_2, \dots, X_6$.

3) Consider that $T$ is simply a statistic of the random sample, and so, from your statistical model for the random vector, you can easily find the statistical model for a function of the random vector. Just remember what you are looking for. A sample space (recall that $T: \mathcal{X}^n\rightarrow \mathcal{T}$, with $\mathcal{T}$ being the space in which the statistic $T$ has values), a probability law (this time for the sum of random geometric variables) and a parameter space.

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