The problem asks to find the Maximum Likelihood Estimator (MLE) for the unknown parameter $\theta$ given a set of independent and identically distributed (i.i.d.) observations $X_1, X_2, \dots, X_n$ from a probability distribution with the probability density function (PDF):

$f_X(x) = \frac{1}{\sqrt{\pi}} \exp \left\{ - (x - \theta)^2 \right\}.$

Step 1: Likelihood Function

The likelihood function $L(\theta)$ for the sample $X_1, X_2, \dots, X_n$ is the product of the individual densities, as the observations are independent. Therefore, the likelihood is:

$L(\theta) = \prod_{i=1}^{n} f_X(x_i) = \prod_{i=1}^{n} \frac{1}{\sqrt{\pi}} \exp \left\{ -(x_i - \theta)^2 \right\}.$

We can simplify this:

$L(\theta) = \left( \frac{1}{\sqrt{\pi}} \right)^n \exp \left\{ - \sum_{i=1}^{n} (x_i - \theta)^2 \right\}.$

Step 2: Log-Likelihood Function

To find the MLE, we typically maximize the log-likelihood function because it is easier to work with sums rather than products. Taking the natural logarithm of the likelihood function gives the log-likelihood $\ell(\theta)$:

$\ell(\theta) = \ln L(\theta) = n \ln \left( \frac{1}{\sqrt{\pi}} \right) - \sum_{i=1}^{n} (x_i - \theta)^2.$

This simplifies further to:

$\ell(\theta) = - n \ln(\sqrt{\pi}) - \sum_{i=1}^{n} (x_i - \theta)^2.$

Step 3: Maximizing the Log-Likelihood

To maximize the log-likelihood, we take the derivative with respect to $\theta$ and set it equal to zero:

$\frac{d}{d\theta} \ell(\theta) = \frac{d}{d\theta} \left( - \sum_{i=1}^{n} (x_i - \theta)^2 \right) = 2 \sum_{i=1}^{n} (x_i - \theta) = 0.$

This simplifies to:

$\sum_{i=1}^{n} (x_i - \theta) = 0.$