My task is to describe the construction of i.i.d. sample $\mathbf{X} = (X_1, \ldots, X_n)$ where random variables $X_1, \ldots, X_n$ should get the predefined univariate distribution $P$. Specifically, it's necessary to describe the "domain" probability space $(\Omega, \mathcal{F}, \mathbf{P})$ and the "induced" probability space $(\mathbb{R}, \mathcal{B}(\mathbb{R}^n), \mathbf{P}_{\mathbf{X}})$ of random vector $\mathbf{X}$ (there is a hint: random vector $\mathbf{X}(\omega)$ should be defined in such a way that the domain measure $\mathbf{P}$ coincides with the induced probability measure $\mathbf{P}_{\mathbf{X}}$.) After that I should also describe the domain and induced probability spaces of each random variable $X_i$.

Please check the correctness of my solution below.

Let's choose $\Omega = \mathbb{R}^n, \, \mathcal{F} = \mathcal{B}(\mathbb{R}^n)$ (Borel sigma-algebra) and

$$\mathbf{P}(B) = \prod_{i=1}^n P(B_i), \quad \forall B = (B_1 \times \ldots \times B_n) \in \mathcal{B}(\mathbb{R}^n).$$

So the domain probability space is $(\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n), \mathbf{P})$. Let's define random vector $\mathbf{X}(\omega) = \omega, ~ \forall \omega \in \mathbb{R}^n$ on this probability space. Then the induced probability space of random vector $\mathbf{X}: \mathbb{R}^n \to \mathbb{R}^n$ is $(\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n), \mathbf{P}_{\mathbf{X}})$ and

$\displaystyle \forall B = (B_1 \times \ldots \times B_n) \in \mathcal{B}(\mathbb{R}^n) \, \hookrightarrow \, \mathbf{P_X}(B) = \mathbf{P}(\mathbf{X} \in B) = \mathbf{P}\{\omega: \mathbf{X}(\omega) \in B\} =$ $= \mathbf{P}\{\omega: \omega \in B \} = \mathbf{P}(B).$

So we have $\mathbf{P_X} = \mathbf{P}$. Therefore, the induced probability space coincides with the domain probability space $(\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n), \mathbf{P})$.

Next, every random variable $X_i(\omega) = \omega_i$ has the same domain probability space as $\mathbf{X}$ (by definition of a random vector) but induces different probability space $(\mathbb{R}, \mathcal{B}(\mathbb{R}), P_{X_i})$ and

$\displaystyle \forall \mathrm{B} \in \mathcal{B}(\mathbb{R}) \, \hookrightarrow \, P_{X_i}(\mathrm{B}) = \mathbf{P}(X_i \in \mathrm{B}) = \mathbf{P}\{\omega: X_i(\omega) \in \mathrm{B} \} = \mathbf{P}\{\omega: \omega_i \in \mathrm{B} \} =$ $ = \mathbf{P}(\mathbb{R} \times \ldots \times \mathbb{R} \times \underbrace{\mathrm{B}}_{i \text{-th place}} \times \mathbb{R} \times \ldots \times \mathbb{R}) = P(\mathbb{R}) \cdots P(\mathbb{R}) \cdot P(\mathrm{B}) \cdot P(\mathbb{R}) \cdots P(\mathbb{R}) = $ $= 1 \cdots 1 \cdot P(\mathrm{B}) \cdot 1 \cdots 1 = P(\mathrm{B}).$

Hence, $P_{X_i} = P$ and the induced probability space of $X_i: \mathbb{R}^n \to \mathbb{R}$ is $(\mathbb{R}, \mathcal{B}(\mathbb{R}), P)$, for all $i$.

P.S. My teacher said that the solution to this problem will show me a typical way of constructing i.i.d. samples in statistics so this is a very important exercise.


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