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My work area is machine learning but since I am not from a math background I am struggling a lot to do the right job.

I wanna define the data distribution which is defined for data points. I have rewritten the following definition many times but my professor is still unhappy with my definition. I am confused if this is the correct definition:

Definition (Distribution): We consider a collection of $T$ data distributions $\mathcal{D}=\{\mathcal{D}_1,\mathcal{D}_2,\dots,\mathcal{D}_T\}$ as the set of $T$ mappings each indexed by $t \in \mathbb{N}$ and denoted as $\mathcal{D}_t(\cdot):\mathbb{R}^{d_t}\times\mathbb{R}^{c_t}\rightarrow\mathbb{R}^+\cup\{0\}$ s.t. $\int_{\mathbb{R}^{d_t}\times\mathbb{R}^{c_t}}\mathcal{D}_t(\cdot)dx^tdy^t=1$ where $(x^t,y^t)\sim\mathcal{D}_t$.

We consider that $t$th distribution is denoted as $\mathcal{D}_t$ from which dataset samples $(x^t_i,y^t_i)$'s are drawn, where $x^t_i \in \mathbb{R}^{d_t}$, $y^t_i \in \mathbb{R}^{c_t}$, and $d_t,c_t\in \mathbb{N}$.

I have three questions:

  1. Is my definition correct? if not, what is the right definition?
  2. Can we have distribution from one data point $(x^t_i,y^t_i)\sim\mathcal{D}_t$? how to write the density function for that?
  3. Is there any book or resources that I can refer to for my math definitions and notations to learn more about the rules for mathematical notations?
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  • $\begingroup$ To me each $\mathcal{D}_t$ is a correctly defined probability density on $\mathbb{R}^{d_t}\times\mathbb{R}^{c_t}$ . For one data point the meaning of $\mathcal{D}_t(x_i^t,y_i)dx_i^tdy_i^t$ is the probability that a random sample lies in a small interval around that point (I would probably omit the sub and superscripts here): $\mathcal{D}_t(x,y)dxdy$ is the same. Any introductory book to probability theory should cover these basic concepts. $\endgroup$
    – Kurt G.
    Sep 21, 2022 at 11:58

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