Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $E[X\ Y^{T}]=E[X]E[Y^{T}]$.

share|improve this question

1 Answer 1

up vote 1 down vote accepted

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 $$ \mathrm E(XY^T)=\int xf_X(x)\mathrm dx\int y^Tf_Y(y)\mathrm dy=\mathrm E(X)\mathrm E(Y^T)=\mathrm E(X)\mathrm E(Y)^T. $$

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.