# A simple question for a Joint Gaussian Random variable. Beginner.

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.

• What makes you think $X,\,Y$ are independent? – J.G. Mar 23 at 13:18
• I have realised my mistake about independence, why does the equality hold though? – 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$$\ln\Bbb E^{sX+tY}=\ln\Bbb E^{(s+ct)X+t(Y-cX)}=\frac{(s+ct)^2+t^2(1-c^2)}{2}=\frac{s^2+t^2+2cst}{2}.$$