Joint density of a bounded random vector

Question

Let $X = (X_1, \dots, X_n)$ be a random vector with support $\mathbb{R}^n$, and with distribution $F_X(x_1, \dots, x_n)$ and density $f_X(x_1,\dots, x_n)$. Consider the bounded transformation of $X$ given by $Y = \min\{\max\{X, a^-\}, a^+\}$, that is, $$ Y = (\min\{\max\{X_1, a^-\}, a^+\}, \dots, \min\{\max\{X_n, a^-\}, a^+\}). $$

I'm trying to find the distribution and density of $Y$, say $F_Y$ and $f_Y$, respectively, but I believe I'm missing something because my results seems contradictory.

Analysing by cases, I got that the distribution of $Y$ takes the form \begin{align*} F_Y(y_1, \dots, y_n) &:= \mathbb{P}(Y_1 \leq y_1, \dots, Y_n \leq y_n) \\ &= \begin{cases} 0, & \text{ if } \exists i : y_i \leq a^- \\ \mathbb{P}\left(\displaystyle{\bigcap_{i\in I}}\{Y_i \leq y_i\}\right), & \text{ if } y_i > a^-\ \forall i,\; \exists i: y_i \leq a^+ \\ 1, & \text{ if } y_i > a^+\ \forall i \end{cases}, \end{align*} where the index set $I$ takes the form $I = \left\{i: a^- < y_i < a^+ \right\}$.

EDIT: Based on the comments I decided to edit this question to address the problem of finding a relatively simple and convenient formula to compute the probabilities $$ \mathbb{P}(Y \in \mathcal{B}), $$ where $\mathcal{B}$ is a as Borel set in the usual topology of $\mathbb{R}^n$.

If $\mathcal{B} = \prod_{i=1}^n\mathcal{B}_i$, where $\{\mathcal{B}_i\}_{i=1}^n$ is a collection of Borel sets in the usual topology in $\mathbb{R}$ such that $\mathcal{B}_i\subset [a^-, a^+]$, then be Borel sets in the , and if we define the sets

\begin{align*} \mathcal{I}_i := \begin{cases} \mathcal{B}_i , & \text{if } a^-\notin \mathcal{B}_i, a^+\notin \mathcal{B}_i \\ \mathcal{B}_i \cup (a^+,\infty), & \text{if } a^-\notin \mathcal{B}_i, a^+\in \mathcal{B}_i \\ \mathcal{B}_i \cup (-\infty, a^-), & \text{if } a^-\in \mathcal{B}_i, a^+\notin \mathcal{B}_i \\ \mathcal{B}_i \cup (-\infty, a^-) \cup (a^+,\infty), & \text{if } a^-\in \mathcal{B}_i, a^+\in \mathcal{B}_i \end{cases}, \end{align*} then, \begin{align*} \mathbb{P}\left\{Y_1 \in \mathcal{B}_1, \dots, Y_n \in \mathcal{B}_n\right\} = \int_{\mathcal{I}_1\times\dots\times\mathcal{I}_n} f_X(x_1,\dots,x_n)\,\mathrm{d} x_1\cdots\mathrm{d} x_n. \end{align*}

However, I cannot express shortly what happens in the case where $\mathcal{B}$ is not the product of borel sets in $\mathbb{R}$. Can you share your thoughts on a general formula to compute such probabilities in the simpler way possible. Many thanks!

Answer 1

In fact, the calculation of the density $f_Y$ is correct. You may use $f_Y$ to calculate the probability of $Y$ lying in a measurable set inside $(a^-,a^+)^n$.

The problem is that the truncated random variable $Y$ is no longer continuous. There is probability mass concentrated on the surface of $[a^-,a^+]^n$, which is of lower dimension.