This is copied from my textbook; Assume that the statistical model for the MVR $\textbf{Y}=(Y_{1},Y_{2},\ldots , Y_{n})^{T}$ is given by the parametric family of joint densities: $$\{ f_Y(y_{1},y_{2},\ldots ,y_{n};\boldsymbol\theta) \}_{\boldsymbol\theta\in\Theta^{k}}$$ with respect to some $\nu_{n}$ in $\mathcal{Y}^{n}$. In the following, the random variable $\textbf{Y}$ will sometimes denote the observations. Assume also that we are given a realization of $\textbf{Y}$ which we shall call the observation set, $\textbf{y}=(y_{1},y_{2},\ldots ,y_{n})^{T}$. We define an estimator as a function $\widehat{\boldsymbol\theta}(\textbf{Y})$.
Now my questions; First of all. The parametric family of joint distributions, are those functions from $\mathbb{R}^{n}$ to $\mathbb{R}$? For example; Let $Y_{1},Y_{2},Y_{3}$ be independent and let $Y_{i}\sim N(\mu, \sigma^2)$ and assume that $\sigma^2=1$. Then the joint density function for $Y_{1},Y_{2},Y_{3}$ is given by $$f_{Y}(\textbf{y};\boldsymbol\mu)=\dfrac{1}{\sqrt{2\pi}}\exp\Bigg (\sum_{k=1}^{3}-\dfrac{(y_{i}-\mu)^2}{2}\Bigg )$$ which for every value of $\mu$ is a function of the three variables $y_{1},y_{2},y_{3}$.
Is this a correct example of the notation ?