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To find the Maximum Likelihood Estimate (MLE) of \(\sigma\), we start with the given probability density function (PDF) of the distribution:

\[
f(x) = \frac{1}{2\sigma} \exp\left(-\frac{|x|}{\sigma}\right)
\]

### Step 1: Set up the likelihood function
Let \(X_1, X_2, \dots, X_n\) be an independent and identically distributed (i.i.d.) sample from this distribution. The likelihood function \(L(\sigma)\) for the sample is the product of the individual PDFs for each data point:

\[
L(\sigma) = \prod_{i=1}^{n} f(X_i) = \prod_{i=1}^{n} \frac{1}{2\sigma} \exp\left(-\frac{|X_i|}{\sigma}\right)
\]

Simplifying this expression:

\[
L(\sigma) = \left(\frac{1}{2\sigma}\right)^n \exp\left(-\frac{1}{\sigma} \sum_{i=1}^{n} |X_i|\right)
\]

### Step 2: Set up the log-likelihood function
The next step is to take the logarithm of the likelihood function, known as the log-likelihood function. This simplifies the product into a sum:

\[
\ell(\sigma) = \log L(\sigma) = n \log \left(\frac{1}{2\sigma}\right) - \frac{1}{\sigma} \sum_{i=1}^{n} |X_i|
\]

Expanding the logarithmic term:

\[
\ell(\sigma) = -n \log(2\sigma) - \frac{1}{\sigma} \sum_{i=1}^{n} |X_i|
\]

### Step 3: Maximize the log-likelihood function
To find the MLE, we differentiate the log-likelihood function with respect to \(\sigma\) and set the derivative equal to zero:

\[
\frac{d}{d\sigma} \ell(\sigma) = -\frac{n}{\sigma} + \frac{1}{\sigma^2} \sum_{i=1}^{n} |X_i| = 0
\]

Multiply through by \(\sigma^2\) to eliminate the denominators:

\[
-n\sigma + \sum_{i=1}^{n} |X_i| = 0
\]

Solve for \(\sigma\):

\[
\sigma = \frac{1}{n} \sum_{i=1}^{n} |X_i|
\]

### Step 4: Interpret the result
Thus, the MLE for \(\sigma\) is the sample mean of the absolute values of the observations:

\[
\hat{\sigma} = \frac{1}{n} \sum_{i=1}^{n} |X_i|
\]

This is the Maximum Likelihood Estimate of \(\sigma\).

---

Would you like further clarification on any part of the process, or would you like to discuss related topics? Here are a few questions to think about:

1. What does the MLE represent in practical terms for the parameter \(\sigma\)?
2. How would you derive the MLE if the sample were not i.i.d.?
3. Can you compare this MLE with the method of moments estimate?
4. How does the nature of the PDF (with absolute value) affect the estimation process?
5. What assumptions do we make about the underlying distribution when deriving MLE?

**Tip:** MLEs are generally consistent and asymptotically normal, meaning they converge to the true parameter value as the sample size increases.

Consider an i.i.d. sample (of size n) of random variables with density function f(x) = (1 / (2σ)) * exp(-|x| / σ) for -∞ < x < ∞. Find the MLE of σ.

Learn how to find the Maximum Likelihood Estimate (MLE) of σ for an i.i.d. sample from a Laplace distribution with PDF f(x) = (1 / (2σ)) * exp(-|x| / σ). This step-by-step solution covers the derivation of the likelihood function and maximization techniques.

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