To obtain the Maximum Likelihood Estimator (MLE) of $p$ for a random sample $X_1, X_2, \ldots, X_n$ from the geometric distribution, we proceed as follows:

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1. Likelihood Function

The probability mass function (PMF) of the geometric distribution is:

$P(X = x) = q^{x-1} p, \quad \text{where } q = 1 - p, \, x = 1, 2, 3, \ldots$

For a random sample $X_1, X_2, \ldots, X_n$, the likelihood function is:

$L(p) = \prod_{i=1}^n P(X_i) = \prod_{i=1}^n q^{X_i - 1} p$

Substitute $q = 1 - p$ into the likelihood function:

$L(p) = \prod_{i=1}^n (1 - p)^{X_i - 1} p$

Simplify:

$L(p) = p^n (1 - p)^{\sum_{i=1}^n (X_i - 1)}$

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2. Log-Likelihood Function

Take the natural logarithm of the likelihood function:

$\ell(p) = \ln L(p) = \ln(p^n) + \ln((1 - p)^{\sum_{i=1}^n (X_i - 1)})$

$\ell(p) = n \ln(p) + \left(\sum_{i=1}^n (X_i - 1)\right) \ln(1 - p)$

Let $S = \sum_{i=1}^n (X_i - 1) = \sum_{i=1}^n X_i - n$. Substitute this:

$\ell(p) = n \ln(p) + S \ln(1 - p)$

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3. Differentiation

Differentiate $\ell(p)$ with respect to $p$:

$\frac{d\ell(p)}{dp} = \frac{n}{p} - \frac{S}{1 - p}$

Set $\frac{d\ell(p)}{dp} = 0$ to find the maximum likelihood estimator of $p$:

$\frac{n}{p} - \frac{S}{1 - p} = 0$

Rearrange:

$\frac{n}{p} = \frac{S}{1 - p}$

Cross-multiply:

$n (1 - p) = S p$

Simplify:

$n - n p = S p$

$n = n p + S p$

$n = p (n + S)$

Solve for $p$:

$\hat{p} = \frac{n}{n + S}$

Substitute $S = \sum_{i=1}^n X_i - n$:

$\hat{p} = \frac{n}{n + \left(\sum_{i=1}^n X_i - n\right)}$

Simplify:

$\hat{p} = \frac{n}{\sum_{i=1}^n X_i}$

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Final Result

The Maximum Likelihood Estimator (MLE) of $p$ is:

$\hat{p} = \frac{n}{\sum_{i=1}^n X_i}$

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Additional Insight:

• The MLE $\hat{p}$ is inversely proportional to the sample mean of $X$, since $\sum_{i=1}^n X_i / n$ is the sample mean.

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Would you like more details, examples, or clarifications on this derivation? 😊

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5 Related Questions:

1. How is the sample mean of $X_1, \ldots, X_n$ related to the MLE of $p$?
2. What is the intuition behind the geometric distribution and its parameters?
3. Can you derive the variance of $\hat{p}$?
4. How does the MLE behave as $n \to \infty$?
5. How would you test the goodness of fit for the estimated geometric model?

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Tip:

Always check if the derived MLE lies in the parameter space (i.e., $0 < p \leq 1$) before using it in applications!