Is the fixed effect in a fixed effect model a random variable or not? From Wikipedia for fixed effect model:

Consider the linear unobserved effects model for N observations and T time periods:
  $$
    y_{it} = X_{it}\mathbf{\beta}+\alpha_{i}+u_{it} \quad t=1,..,T, \quad i=1,...,N
$$
  where $y_{it}$ is the dependent variable observed for individual $i$ at time $t$, $X_{it}$ is the time-variant $1\times k$ regressor matrix, $\alpha_{i}$ is the unobserved time-invariant individual effect and $u_{it}$ is the error term. Unlike $X_{it}$, $\alpha_{i}$ cannot be observed by the econometrician. Common examples for time-invariant effects $\alpha_{i}$ are innate ability for individuals or historical and institutional factors for countries.
Unlike the Random effects (RE) model where the unobserved $\alpha_{i}$ is independent of $x_{it}$ for all $t=1,...,T$, the FE model allows $\alpha_{i}$ to be correlated with the regressor matrix $x_{it}$. Strict exogeneity, however, is still required.
Since $\alpha_{i}$ is not observable, it cannot be directly controlled for. The FE model eliminates $\alpha_{i}$ by demeaning the variables using the within transformation:
  $$
    y_{it}-\overline{y_{i}}=\left(X_{it}-\overline{X_{i}}\right) \beta+ \left( \alpha_{i} - \overline{\alpha_{i}} \right ) + \left( u_{it}-\overline{u_{i}}\right) = \ddot{y_{it}}=\ddot{X_{it}} \beta+\ddot{u_{it}}
$$
  where $\overline{X_{i}}=\frac{1}{T}\sum\limits_{t=1}^{T}X_{it}$ and $\overline{u_{i}}=\frac{1}{T}\sum\limits_{t=1}^{T}u_{it}$. Since $\alpha_{i}$ is constant, $\overline{\alpha_{i}}=\alpha_{i} $ and hence the effect is eliminated. The FE estimator $\hat{\beta}_{FE}$ is then obtained by an OLS regression of $\ddot{y}$ on $\ddot{X}$.

On one hand, it says "the FE model allows $\alpha_{i}$ to be correlated with the regressor matrix $x_{it}$", which I understand to be that both $\alpha_{i}$ and $x_{it}$ are random variables. On the other hand, it says "$\alpha_{i}$ is constant", and my previous impression is that  $\alpha_{i}$ is nonrandom in fixed effect model, and random in random effect model. So I was wondering how to correctly understand "fixed effect"?
Thanks and regards!
 A: My answer would be that $\alpha_i$, usually interpreted as the unobserved heterogeneity, is a random variable in the sense that it varies over individuals. In that sense, you can say that it maybe correlated with other variables. However in the fixed effect model, $\alpha_i$ is not treated by the model as a random variable. It's just a set of parameters and in a frequentist framework, parameters are not considered as random variables. 
I'm not a specialist in bayesian statistics, but I can imagine that in a bayesian framework, you would not have this "puzzle" since parameters are themselves treated as random variables. 
A: I think the main difference between fixed and random effects is that the unobserved individual effect $\alpha_i$ is purely random in the random effect paradigm in the sense that its distribution does not depend on the observed $X_{it}$; under the fixed effect settings, the distribution of $\alpha_i$ depends on $X_{it}$. The fixed effect $\alpha_i$ became a constant after sampling. For example, let $\alpha_i$ be one's ability. The ability $\alpha_i$ among the population is surely a random variable regardless of the dependence of its distribution upon $X_{it}$. Given sampled data, the ability, $\alpha_i$, of a particular individual $i$ is constant, and the fixed effect estimator applies. 
