Question 1164708
To solve this problem, we first identify the distribution described. The density function you provided is:

This is the probability density function of a **Gamma Distribution**, denoted as .

### 1. Key Properties of the Distribution

For a single random variable  following this distribution, the mean and variance are defined as:

* **Mean ():** 
* **Variance ():** 

### 2. Properties of the Random Sample

Since you have a random sample of size  (), we can determine the distribution of the **sample sum** and the **sample mean**.

#### A. Distribution of the Sum ()

The sum of independent Gamma variables with the same scale parameter  also follows a Gamma distribution.

* **New Alpha:** 
* **Scale:**  remains the same.



#### B. Distribution of the Sample Mean ()

By the **Central Limit Theorem**, since your sample size () is sufficiently large (typically ), the sampling distribution of the mean will be approximately **Normal**, regardless of the underlying Gamma shape.

The parameters for this normal distribution would be:

* **Mean of :** 
* **Variance of :** 

### 3. Moment Generating Function (MGF)

The MGF for a single  is:


For the entire sample sum :


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**Would you like me to calculate a specific probability for this sample, or find the maximum likelihood estimators (MLE) for  and ?**