$X$, $Y$ independent if and only if $X$, $Y$ uncorrelated.

Question

Suppose that the joint probability density function of $(X, Y)$ is given by $f_{X,Y}(x, y) =[1 - \alpha(l-2x)(l-2y)]I_{(o,1)}(x)I_{(o,1)}(x)$ where -1 < $\alpha$ < 1.

Prove or disprove: $X$ and $Y$ are independent if and only if $X$ and $Y$ are uncorrelated.

Answer 1

Solution:

First, let's find the marginal distribution for $X,Y$

$f_X(x)=\int_{0}^{1}[1 - \alpha(l-2x)(l-2y)]I_{(o,1)}(x)dy=I_{(o,1)}(x)$

$f_Y(y)=\int_{0}^{1}[1 - \alpha(l-2x)(l-2y)]I_{(o,1)}(y)dx=I_{(o,1)}(y)$

Then, let's verify that $E[XY]=E[X]E[Y]$

$E[XY]=\iint xI_{(o,1)}(x)yI_{(o,1)}(y)dxdy=[\int xI_{(o,1)}(x)][\int yI_{(o,1)}(y)]=E[X]E[Y]$.

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Given the second step, $X,Y$ are uncorrelated even if they are not independent.

The statement if false then.