I am a beginner in probability theory.

The Wikipedia definition of joint Gaussian random variables is https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Notation_and_parametrization it says that the random variables have to be independent...

Let $X,Y$ be jointly normal with $\mathbb{E}(X)=\mathbb{E}(Y)=0$, and $\mathbb{E}(X^2)=\mathbb{E}(Y^2)=1 $, I am reading a probability book where it says

$$ \mathbb{E}(e^{sX-\frac{s^2}{2}}e^{tY-\frac{t^2}{2}})=e^{st\mathbb{E}(XY)}.$$

Can anyone explain the equality.

  • $\begingroup$ What makes you think $X,\,Y$ are independent? $\endgroup$ – J.G. Mar 23 at 13:18
  • $\begingroup$ I have realised my mistake about independence, why does the equality hold though? $\endgroup$ – joedondonjoe Mar 23 at 13:30

Suppose $X,\,Y$ have mean $0$ and variance $1$. Since $X$ is uncorrelated with and hence independent of $Z:=Y-cX$ with $c:=\operatorname{Cov}X,\,Y=\Bbb EXY$, and since $Z$ has mean 0 and variance$$\operatorname{Var}Y+c^2\operatorname{Var}X-2c\operatorname{Cov}(X,\,Y)=1-c^2,$$it follows that$$\ln\Bbb E^{sX+tY}=\ln\Bbb E^{(s+ct)X+t(Y-cX)}=\frac{(s+ct)^2+t^2(1-c^2)}{2}=\frac{s^2+t^2+2cst}{2}.$$

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