rigorous definition of maximum likelihood estimation I am reading the Pattern Recognition and Machine Learning, written by Bishop. On page 27, the parameter of Guassian distribution $\mu$ can be regarded as 
$$\mu_{\text{ML}}=\frac{1}{N}\sum_1^N x_n$$
The $x_n$ are observed value, and they are real. However, author also see them as a random variables. Is there any probability theory to explain the contradiction.
Any advice is helpful. Thank you.
 A: First, it's better to write $\mu_{\text{ML}}$ as $\hat{\mu}_{\text{ML}}$ to highlight it is an estimator rather a parameter --- this is a convention used extensively in statistics (in the following I will denote it by $\hat{\mu}$ for simplicity). 
To answer your question, depending on the purposes of analysis, $\hat{\mu}$ can be treated as either random variable (function) or observed value (real number).
If the purpose is to explore the statistical/mathematical properties, such as compute its mean/variance, derive its asymptotic distribution, etc, we should treat $\hat{\mu}$ as a random variable, under which case it's better to use capital case lettersto denote the sample and write
$$\hat{\mu} = \frac{1}{N}\sum_{n = 1}^N X_n,$$
and in some literature $\hat{\mu}$ is called an estimator in that it is its distributional property instead of its numerical value is of our primary interest. This perspective is mostly taken in mathematical/theoretical statistics course/literature.
If the purpose is to analyze a particular data set and draw some empirical conclusion, $\hat{\mu}$ is treated as observed value, under which case it's more conventional to denote sample by lower case letters and write
$$\hat{\mu} = \frac{1}{N}\sum_{n = 1}^N x_n,$$
and sometimes $\hat{\mu}$ is referred as an estimate. This perspective is mostly taken in applied statistics and data mining.
The underlying reason for such distinction lies in whether the sample $S = \{X_1, X_2, \ldots, X_N\}$ has been observed/obtained. Before the observation/experimentation has been made to $S$, the only sensible way to deal with $S$ is to view it as a collection of random variables, consequently, all we can do with it and its function (such as $\hat{\mu}$ in this example) is to discuss mathematical properties under some assumptions. If the observation/experimentation has been made to $S$, the numerical values of $S$ (called realization of $S$) is available to us, under which we can conduct any reasonable data analysis and make statistical inference (which, of course, relies on the theoretical properties of its corresponding estimator). 
