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The text in the image is in Chinese and appears to ask for the **maximum likelihood estimation (MLE)** for the parameters \(\alpha\) and \(\lambda\) of the Gamma distribution, denoted as \(Ga(\alpha, \lambda)\), where \(\alpha > 0\) and \(\lambda > 0\).

### Translation of the problem:
"Assume the sample \(X_1, X_2, \dots, X_n\) comes from a Gamma distribution \(Ga(\alpha, \lambda)\), where \(\alpha > 0\) and \(\lambda > 0\). Provide the estimation of parameters \(\alpha\) and \(\lambda\)."

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### Solution Strategy:

For a Gamma distribution \(X \sim Ga(\alpha, \lambda)\), with probability density function (PDF):
\[
f(x; \alpha, \lambda) = \frac{\lambda^\alpha x^{\alpha - 1} e^{-\lambda x}}{\Gamma(\alpha)}
\]
where \(\Gamma(\alpha)\) is the Gamma function, and \(\alpha, \lambda\) are the shape and rate parameters, respectively.

We will use Maximum Likelihood Estimation (MLE) to estimate \(\alpha\) and \(\lambda\).

#### 1. **Log-likelihood function:**
Given a sample \(X_1, X_2, \dots, X_n\), the likelihood function is:
\[
L(\alpha, \lambda) = \prod_{i=1}^n \frac{\lambda^\alpha X_i^{\alpha - 1} e^{-\lambda X_i}}{\Gamma(\alpha)}
\]
Taking the logarithm:
\[
\log L(\alpha, \lambda) = n\alpha \log \lambda - n \log \Gamma(\alpha) + (\alpha - 1) \sum_{i=1}^n \log X_i - \lambda \sum_{i=1}^n X_i
\]

#### 2. **Partial derivatives and MLE equations:**
We compute the partial derivatives with respect to \(\alpha\) and \(\lambda\), and set them to zero to solve for the parameters.

- **Derivative with respect to \(\lambda\):**
\[
\frac{\partial \log L}{\partial \lambda} = \frac{n\alpha}{\lambda} - \sum_{i=1}^n X_i = 0
\]
Solving this gives the MLE for \(\lambda\):
\[
\hat{\lambda} = \frac{n\alpha}{\sum_{i=1}^n X_i}
\]

- **Derivative with respect to \(\alpha\):**
\[
\frac{\partial \log L}{\partial \alpha} = n \log \lambda - n \frac{\Gamma'(\alpha)}{\Gamma(\alpha)} + \sum_{i=1}^n \log X_i = 0
\]
The expression \(\frac{\Gamma'(\alpha)}{\Gamma(\alpha)}\) is known as the digamma function, \(\psi(\alpha)\).

This equation can be solved numerically for \(\alpha\), using numerical methods like Newton-Raphson.

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Would you like a more detailed walkthrough of the numerical solution for \(\alpha\), or any specific step in the derivation? Let me know if you need more details!

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**Further Exploration Questions:**

1. What is the significance of the Gamma distribution in statistical modeling?
2. How does the Newton-Raphson method work for solving MLE equations?
3. What is the role of the digamma function \(\psi(\alpha)\) in this estimation process?
4. Could you use alternative methods, such as the Method of Moments, to estimate \(\alpha\) and \(\lambda\)?
5. How would you extend this estimation technique to a multivariate Gamma distribution?

**Tip:** For Gamma distribution MLE, the equation for \(\alpha\) often requires iterative methods since it involves special functions (like the digamma function). Numerical solvers like `scipy` in Python are commonly used.

Assume the sample X_1, X_2, ..., X_n comes from a Gamma distribution Ga(α, λ), where α > 0 and λ > 0. Provide the estimation of parameters α and λ.

This page explains the Maximum Likelihood Estimation (MLE) for the parameters α and λ of a Gamma distribution. Learn about the log-likelihood function, how to derive the MLE equations, and how to solve for these parameters numerically. Suitable for graduate-level studies in statistics.

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