# Independent random variables.

We define probability distributions $F(x,y)=P(X\leq x, Y\leq y)$ where X,Y are two arrays of random variables, and probability density function $\frac{\partial^{n+m} F(x,y)}{\partial x_{1}...\partial x_{n}\partial y_{1}...\partial y_{m}}$, where $X=(X_{1}...X_{n}), Y=(Y_{1}...Y_{m})$. X and Y are independent if and only if $F(x,y)=F(x)F(y), f(x,y)=f(x)f(y)$. There is no linear correlation between X ad Y if $E[X\ Y^{T}]=E[X]E[Y^{T}]$. I want to prove that, if X and Y are independent then $E[X\ Y^{T}]=E[X]E[Y^{T}]$.

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By definition of expectation and independence, $$\mathrm E(XY^T)=\iint xy^Tf_{X,Y}(x,y)\mathrm dx\mathrm dy=\iint xy^Tf_X(x)f_Y(y)\mathrm dx\mathrm dy.$$ Fubini theorem yields $$\mathrm E(XY^T)=\int xf_X(x)\mathrm dx\int y^Tf_Y(y)\mathrm dy=\mathrm E(X)\mathrm E(Y^T)=\mathrm E(X)\mathrm E(Y)^T.$$