# Notation for a derived probability measure

Suppose that I have a random variable $$X$$, which is a variable-dimensional (i.e. it could be 1 dimensional, 2 dimensional, 3 dimensional, etc.). The dimension of a specific $$x$$ is known and denoted as $$d(x)$$. The distribution of $$X$$ can be discrete, continuous, or mixed, and I use the measure $$\mathbb P_X$$ to refer to it.

Now, I am interested in the indexed entry $$Y=(X, I)$$, where $$1\leq I \leq d(X)$$. In other words, each $$Y$$ is a joint vector containing the full vector $$X$$ and an index $$I$$. Suppose I know that all the $$Y$$s corresponding to the same $$X$$ have uniform distribution (i.e. with probability $$1/d(X)$$). Thus, I can sample $$Y$$ from the following procedure:

1. $$X\sim \mathbb P_X$$
2. $$I\sim \mathrm{Unif}(\{1, ..., d(X)\})$$
3. $$Y=(X, I)$$

What is the formal way of writing down the general probability measure notation for $$Y$$? Specifically, I want to express $$\mathbb P_Y = ?$$. The tricky part is that whether $$Y$$ is continuous or discrete depends on that of $$X$$, and ideally I want a unified notation to cover both cases (and the case of hybrid distribution). I am thinking about something like $$\mathbb P_Y=\frac{1}{d(X)}\mathbb P_X$$, but this doesn't seem correct as it has an "undefined" variable $$X$$ and I don't even think I can divide a probability measure by something.

• At the moment your value $Y=(X,I)$ is not just a specific entry of $X$ --- it is the joint vector containing $X$ and also a random index value. Do you perhaps intend to have $Y=X_I$ instead, or is your description of this variable wrong?
– Ben
Aug 23, 2021 at 6:22
• @Ben Oh sorry for the confusion. I did intend it to be $Y=(X, I)$, which is the joint distribution of $X$ and a random index value of it.
– Y.Z.
Aug 23, 2021 at 23:35
• Okay; in view of that I have made an edit to reword your description of $Y$. Please check that this accords with your understanding of the question.
– Ben
Aug 24, 2021 at 1:21

In order to stress that some of your objects are vectors, I am going to call them $$\mathbf{Y}$$ and $$\mathbf{X}$$ in bold notation.

We usually don't try to write probability measures explicitly, since they are functions defined on a sigma-field, and this makes them cumbersome to write in explicit form. When we want to set out explicit results for a situation where we may have discrete or continuous (or mixed) random variables, we usually do this using the cumulative distribution function.

Simplifying the distribution: In your case, the main challenge here is dealing with a random vector with variable dimension. A simple and effective way to deal with this is to define this random vector using two parts: a sequence giving all elements when the dimension is infinite; and a random variable determining the actual dimension. To do this, suppose we define a random sequence $$\mathbf{X}_\infty = (X_1,X_2,X_3,...)$$, a random dimension $$D$$ and a random index $$I$$ and we then define:

\begin{align} \mathbf{X} &\equiv (X_1,...,X_D), \\[6pt] \mathbf{Y} &\equiv (X_1,...,X_D,I). \\[6pt] \end{align}

The random object $$(\mathbf{X}_\infty, D)$$ fully determines $$\mathbf{X}$$ and the random object $$(\mathbf{X}_\infty, D, I)$$ fully determines $$\mathbf{Y}$$. We can characterise the distribution of interest by noting the conditional independence $$I \ \bot \ \mathbf{X}_\infty | D$$ and then decomposing it into the following parts:

\begin{align} p(d) &\equiv \mathbb{P}(D=d), \\[10pt] p_d(i) &\equiv \mathbb{P}(I=i |D=d) = \frac{\mathbb{I}(i \in \{1,...,d\})}{d}, \\[12pt] F_d(\mathbf{x}_\infty) &\equiv \mathbb{P}(\mathbf{X}_\infty \leqslant \mathbf{x}_\infty|D=d). \\[6pt] \end{align}

This gives us some structure that makes it simpler to write the joint distribution of interest, leading to the probability measure of interest.

Writing the distribution of interest: Now, let $$\mathbf{y} = (\mathbf{x}, i) = (x_1,...,x_d, i)$$ denote a specific value for the outcome $$\mathbf{Y}$$. Regardless of whether the values in $$\mathbf{X}_\infty$$ are discrete or continuous (or mixed) we can encompass all information of interest using the following function:

\begin{align} H_\mathbf{Y}(\mathbf{y}) &\equiv \mathbb{P}(\mathbf{X} \leqslant \mathbf{x}, I = i) \\[6pt] &= \mathbb{P}(\mathbf{X} \leqslant \mathbf{x}, I = i, D = d) \\[6pt] &= \mathbb{P}(\mathbf{X} \leqslant \mathbf{x}| I = i, D = d) \cdot p_d(i) \cdot p(d) \\[6pt] &= F_d((\mathbf{x},\infty,\infty,\infty,...)) \cdot p_d(i) \cdot p(d). \\[6pt] \end{align}

This function is a mixture of a CDF (with respect to $$\mathbf{X}$$) and a mass function (with respect to $$I$$). It uniquely determines the probability measure $$\mathbb{P}_\mathbf{Y}$$. Specifically, this function allows you to define the pre-measure:

$$\mathbb{P}_\mathbf{Y}((-\infty, \mathbf{x}] \times (-\infty,i]) = \sum_{j=1}^{\lfloor i \rfloor} H_\mathbf{Y}((\mathbf{x},j)),$$

and the full probability measure is then uniquely defined using the Carathéodory extension theorem.