Question 1164708: Let X1, X2, X3, ...,X 50 be a random sample of size 50 from a distribution
with density function given by:
f(x) = product of x^(a-1) and exp^(x/Q) divided by product of gamma of alpha
and Q^(a) with the condition 0 < x < inf. [Here a means alpha]
Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website! 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 ?**
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