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THEOREM: Let random variables $X$ and $Y$ have a continuous joint distribution. Suppose that $\{(x, y) :f(x, y) > 0\}$, (where $f(x,y)$ is a p.f. or p.d.f. or p.f./p.d.f.) is a rectangular region $R$ (possibly unbounded) with sides (if any) parallel to the coordinate axes. Then $X$ and $Y$ are independent if and only if $f(x,y)=h_1(x)\cdot h_2(y)$ holds for all $(x, y) ∈ R$, where $h_1$ and $h_2$ are nonnegative functions.

MY TRY:

Take function $f_1(x)$ as defined as marginal p.f./p.d.f of $X$ for the region $R$ and outside $0$.Similarly take function $f_2(x)$ as defined as marginal p.f./p.d.f of $Y$ for the region $R$ and outside $0$.

But I don't think my way is complete and beautiful. Please let me also know what if sides are not parallel to the coordinate axes or are curved? In such cases is it always the case that $X$ and $Y$ are dependent?

Silent
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1 Answers1

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$X$ and $Y$ are independent if and only if $P(X\land Y)=P(X)P(Y)$.

Suppose that $X$ and $Y$ are independent. Define $$ h_1(x)=\int_{E_Y}f(x,y)\,\mathrm{d}y\tag{1} $$ and $$ h_2(y)=\int_{E_X} f(x,y)\,\mathrm{d}x\tag{2} $$ The probability that $x\in A\subset E_X$ is $$ \int_Ah_1(x)\,\mathrm{d}x=\int_A\int_{E_Y}f(x,y)\,\mathrm{d}y\,\mathrm{d}x\tag{3} $$ The probability that $y\in B\subset E_Y$ is $$ \int_Bh_2(y)\,\mathrm{d}y=\int_B\int_{E_X}f(x,y)\,\mathrm{d}x\,\mathrm{d}y\tag{4} $$ By independence The probability that $x\in A$ and $y\in B$ is $$ \int_B\int_Af(x,y)\,\mathrm{d}x\,\mathrm{d}y=\int_Ah_1(x)\,\mathrm{d}x\int_Bh_2(y)\,\mathrm{d}y\tag{5} $$ Let $A\times B$ be a small rectangular region about some point $(x_0,y_0)$. Then, since $f$ is continuous, we get from $(5)$ that $$ f(x_0,y_0)=h_1(x_0)h_2(y_0)\tag{6} $$ The other direction is even simpler (that is, given $f(x,y)=h_1(x)h_2(y)$, show $X$ and $Y$ are independent).

robjohn
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