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The problem asks to show that the given family of distributions is an **exponential family** (ΜΕΟΚ in Greek, short for "Μέγιστη Εκθετική Οικογένεια Κατανομών") and to find a sufficient statistic for this family. The given probability density function is:

\[
f_1(x; \theta) = \frac{3\theta x^2}{(1 + x^3)^{\theta + 1}} \mathbb{I}_{(0, \infty)}(x), \quad \theta \in \Theta = (0, \infty).
\]

### Step 1: Show that the family of distributions belongs to the exponential family

To show that a given family belongs to the exponential family, we express the density function in the following general form:

\[
f(x; \theta) = h(x) \exp \left( \eta(\theta) T(x) - A(\theta) \right),
\]

where:
- \(h(x)\) is a function of the data,
- \(\eta(\theta)\) is the natural parameter,
- \(T(x)\) is the sufficient statistic,
- \(A(\theta)\) is the log-partition function (or cumulant function).

We start by rewriting the given density:

\[
f_1(x; \theta) = \frac{3 \theta x^2}{(1 + x^3)^{\theta + 1}}.
\]

We can separate terms involving \(x\) and \(\theta\):

\[
f_1(x; \theta) = 3 x^2 \cdot \theta \cdot \left( \frac{1}{(1 + x^3)} \right)^{\theta + 1}.
\]

Take the logarithm of the function to help express it in exponential form:

\[
\log f_1(x; \theta) = \log(3 x^2) + \log(\theta) - (\theta + 1)\log(1 + x^3).
\]

This can be rearranged as:

\[
\log f_1(x; \theta) = \log(3 x^2) - \log(1 + x^3) + \theta \left( -\log(1 + x^3) + \log(\theta) \right).
\]

Comparing this with the form of the exponential family:

- The term \(\log(3 x^2) - \log(1 + x^3)\) is independent of \(\theta\), so it corresponds to \(h(x)\).
- The term \(-\log(1 + x^3)\) can be treated as \(T(x)\), which is the sufficient statistic.
- \(\eta(\theta) = \log(\theta)\) is the natural parameter.
- \(A(\theta)\) corresponds to the terms related to normalization.

Thus, the family is part of the exponential family.

### Step 2: Find a sufficient statistic

For the exponential family, the sufficient statistic is typically the function of \(x\) that multiplies the natural parameter. In this case, the sufficient statistic \(T(x)\) is:

\[
T(x) = -\log(1 + x^3).
\]

### Final Answer:

1. The family of distributions belongs to the exponential family.
2. A sufficient statistic for this family is \(T(x) = -\log(1 + x^3)\).

Let me know if you'd like a deeper explanation or have more questions! Here are five related questions:

1. What are the properties of a sufficient statistic in the exponential family?
2. How do you compute the maximum likelihood estimator for \(\theta\) given this family?
3. Can you derive the cumulant function \(A(\theta)\) for this specific exponential family?
4. What are the advantages of recognizing a distribution as part of the exponential family?
5. How does the concept of conjugate priors apply to distributions from the exponential family?

**Tip**: Understanding how to express a probability density function in the exponential family form can simplify solving problems related to sufficient statistics and maximum likelihood estimation.

Show that the family of distributions is an exponential family and find a sufficient statistic for the given distribution.

This problem demonstrates how to prove that a family of distributions belongs to the exponential family and identifies the sufficient statistic. It uses key statistical concepts like the exponential family and the factorization theorem.

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