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Consider the random variables $X$ and $Y$ with joint distribution given by

enter image description here

We know that the for the random vector $X=(X_{1}, \dots, X_{n})$, for $j \in \{1, \dots, n \}$

$$P(X_{j}=x)=\sum_{x_{1}}\sum_{x_{2}} \cdots \sum_{x_{n}}P(X_{1}=x_{1}, \dots, X_{n}=x_{n})$$

For the example we have that the marginal of $X$ and $Y$ is

enter image description here

enter image description here

But I want to know what the sums look like, e.g.

$$P(X=0)=\sum_{\text{what here?}} P(\text{what here?})$$

I don't see how obvious it is to go from definition to example, any help for this?

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    $\begingroup$ $P(X=0)=\frac18+0+0+0= \frac18$. $P(X=1)=0 +\frac18 +\frac18=\frac38$. Just add probabilities across the appropriate row (and write in the margin). Similarly columns for $Y$. $\endgroup$
    – Henry
    Commented Nov 26, 2023 at 2:27

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