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I know that for a random variable $X$ the empirical distribution using $N$ samples of $X$ can be found by the following:

\begin{equation*} \nu=\frac{1}{N}\sum_{i=1}^{N}\delta_{\hat{X}^{(i)}} \end{equation*}

where $\delta_{\hat{X}^{(i)}}$ is the Dirac delta measure concentrated at $\hat{X}^{(i)}$.

However, if I have two random variables $X$ and $Y$, how can I find the empirical joint distribution given $N$ samples of $X$ and $Y$?

Thank you!

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The empirical (probability) measure in $\mathbb{R}^d$ is defined in the same way. For $A\in \mathcal{B}(\mathbb{R}^d)$, $$ \nu_n(A)=\frac{1}{n}\sum_{i=1}^n 1\{X_i\in A\}. $$

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