Definition

statistical model

We are given a random experiment with sample space $\Omega$. On this sample space we have defined a random vector $\mathbf{X} = (X_1 , \dots , X_n )$. When $\omega$ is the outcome of the experiment, $\mathbf{X}(\omega)$ is referred to as the observations or data. It is often convenient to identify the random vector $\mathbf{X}$ with its realization, the data $\mathbf{X}(\omega)$. Since it is only $\mathbf{X}$ that we observe, we need only consider its probability distribution. This distribution is assumed to be a member of a family $\mathcal{P}$ of probability distributions on $\mathbb{R}^{n}$. $\mathcal{P}$ is refered to as the model.