The image contains three mathematical problems, all dealing with statistics and sufficiency. Let's break them down one by one.

Problem 1:

Given:
A random sample $X_1, X_2, \dots, X_n$ is drawn from a distribution with the PDF: $f(x; \theta) = \theta e^{-\theta x}, \quad 0 < x < \infty, \theta > 0$ Show that the statistic $S = \sum_{i=1}^{n} X_i$ is a sufficient statistic for $\theta$.

Solution Outline: This problem involves proving sufficiency using the Factorization Theorem. According to this theorem, a statistic $T(X)$ is sufficient for $\theta$ if the likelihood function $L(\theta; X_1, X_2, \dots, X_n)$ can be factorized into two parts:

1. One part depends on the data $X_1, X_2, \dots, X_n$ only through the statistic $T(X)$,
2. The other part does not depend on $\theta$.

Problem 2:

Given:
A random sample of size $n$ is drawn from a geometric distribution, $X_i \sim \text{Geo}(p)$. Show that the statistic $S = \sum_{i=1}^{n} X_i$ is a sufficient statistic for $p$. The probability mass function of $X \sim \text{Geo}(p)$ is given by: $f(x) = p(1 - p)^{x - 1}, \quad x = 1, 2, \dots, \quad 0 < p < 1$

Solution Outline: Similar to the first problem, we will apply the Factorization Theorem to show that the statistic $S = \sum_{i=1}^{n} X_i$ is sufficient for $p$.

Problem 3:

Given:
A random sample $X_1, X_2, \dots, X_n$ is drawn from $X_i \sim N(0, \theta)$. Show that the statistic $S = \sum_{i=1}^{n} X_i^2$ is a sufficient statistic for $\theta$.

Solution Outline: Here, $X_i$ are distributed according to a normal distribution with mean 0 and variance $\theta$. To show sufficiency, we will use the Factorization Theorem as in the previous two problems.

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Here are 5 related questions:

1. What is the Factorization Theorem and how does it help in determining sufficient statistics?
2. How can we apply the Factorization Theorem to a Poisson distribution?
3. How would sufficiency differ if we used a Bayesian approach?
4. What are minimal sufficient statistics and how do they relate to sufficient statistics?
5. How does sufficiency help in estimating parameters in hypothesis testing?

Tip: The Factorization Theorem is the most common tool used to determine sufficiency of statistics and is essential in simplifying statistical inference.