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Given the pdfs of two discrete independent variables $X$ and $Y$, write the joint pdf. There is a property that $ if\ \ p_{XY}(x,y) = p_X(x)p_Y(y) \ \forall i,j \Rightarrow \text{X,Y are independent}$. I am not that sure if this statement is true for $"\Leftarrow"$ and I cannot find another way to solve this problem.

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The converse is also true and you can write the joint pmf as the multiplication of the two marginal pmfs.

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