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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.

The image contains three mathematical problems: 1. A random sample from an exponential distribution, show that the sum of the sample is sufficient for θ. 2. A random sample from a geometric distribution, show that the sum of the sample is sufficient for p. 3. A random sample from a normal distribution N(0, θ), show that the sum of the squares of the sample is sufficient for θ.

Explore how to prove the sufficiency of statistics using the Factorization Theorem for exponential, geometric, and normal distributions. This detailed guide covers the necessary steps to show that the sum of samples is sufficient for parameters such as θ and p.

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