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Let's tackle the three parts of Problem 1 step by step. We're dealing with a geometric distribution where \( X \) follows the probability mass function:

\[
P(X = k) = p(1 - p)^{k-1}
\]

Given this, we can proceed with the three parts:

### Part (a): Method of Moments Estimator for \( p \)

For a geometric distribution, the expected value (mean) is given by:

\[
E[X] = \frac{1}{p}
\]

The method of moments involves equating the population mean \( E[X] \) to the sample mean \( \bar{X} \). Hence, we set:

\[
\bar{X} = \frac{1}{p}
\]

Solving for \( p \), we get the method of moments estimator:

\[
\hat{p}_{MM} = \frac{1}{\bar{X}}
\]

### Part (b): Maximum Likelihood Estimator (MLE) for \( p \)

The likelihood function for a random sample \( X_1, X_2, \dots, X_n \) from a geometric distribution is:

\[
L(p) = \prod_{i=1}^{n} p(1 - p)^{x_i - 1}
\]

The log-likelihood function is:

\[
\ln L(p) = \sum_{i=1}^{n} \left( \ln p + (x_i - 1) \ln(1 - p) \right)
\]

Simplifying, we get:

\[
\ln L(p) = n \ln p + \sum_{i=1}^{n} (x_i - 1) \ln(1 - p)
\]

To find the MLE, we take the derivative of \( \ln L(p) \) with respect to \( p \) and set it to zero:

\[
\frac{d}{dp} \ln L(p) = \frac{n}{p} - \frac{\sum_{i=1}^{n} (x_i - 1)}{1 - p} = 0
\]

Solving for \( p \), we get:

\[
\hat{p}_{MLE} = \frac{n}{\sum_{i=1}^{n} x_i}
\]

### Part (c): Maximum Likelihood Estimate for \( p \) with the Given Sample

The sample data is \( \{1, 5, 1, 2, 6, 2, 3, 7, 3, 8\} \). To find the MLE, we first compute the sum of the sample values:

\[
\sum_{i=1}^{n} x_i = 1 + 5 + 1 + 2 + 6 + 2 + 3 + 7 + 3 + 8 = 38
\]

With \( n = 10 \) (since there are 10 values in the sample), the MLE for \( p \) is:

\[
\hat{p}_{MLE} = \frac{10}{38} = \frac{5}{19} \approx 0.2632
\]

### Summary:
- **Method of Moments Estimator**: \( \hat{p}_{MM} = \frac{1}{\bar{X}} \)
- **MLE for \( p \)**: \( \hat{p}_{MLE} = \frac{n}{\sum x_i} \)
- **MLE from the given sample**: \( \hat{p}_{MLE} = \frac{5}{19} \approx 0.2632 \)

Would you like more detailed steps on any part of this solution?

Here are 5 related questions to expand on this:
1. How does the method of moments differ from the maximum likelihood estimation?
2. How would you calculate the variance of the geometric distribution?
3. How does the shape of the geometric distribution change as \( p \) increases?
4. Can you apply the same estimators for other discrete distributions like the binomial or Poisson distributions?
5. How can we check the consistency of the estimators for large sample sizes?

**Tip**: Maximum likelihood estimators are asymptotically efficient, meaning as the sample size increases, they become the best possible estimators with the least variance.

Suppose that X follows a geometric distribution: P(X = k) = p(1 - p)^(k-1) and assume a random sample X1, X2, ..., Xn. a) Find the method of moments estimator of p. b) Find the MLE of p. c) Find the maximum likelihood estimate of p if the collected sample dataset is {1, 5, 1, 2, 6, 2, 3, 7, 3, 8}.

Learn how to find the method of moments estimator and the maximum likelihood estimator (MLE) for a geometric distribution. Example includes finding the MLE for a sample dataset {1, 5, 1, 2, 6, 2, 3, 7, 3, 8}.

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