# Using MGF for multivariate random variables

How do you use MGF for solving moment based questions for multivariate random variables?

For the single variable case, we:

find $E(e^{tX})$, find the interval in which it exists (around 0), differentiate it n times and plug in t as 0 to receive the n'th moment $E(X^n)$.

With a general multivariate distribution like $f_{XY}(x,y)$, how do I use MGF to find stuff like $E(X), E(XY), E(Y)$ etc?

I read wikipedia and some other pages but they keep losing me at some point.

I understand the MGF will be $E(e^{t'X})$ where t and X are now vectors but where do I go from here?

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Let's write down what your lasyt expression means in dimension $2$: $$m(t_1,t_2) = E[e^{t_1X_1 + t_2 X_2}],$$ then we get $$\frac{\partial m}{\partial t_i} = E[X_i e^{t_1X_1 + t_2 X_2}]$$ and we can evaluate at $t_1=t_2 = 0$ to get $E[X_i]$ for $i = 1,2$. More interesting: $$\frac{\partial m^2}{\partial t_1 \partial t_2} = E[X_1 X_2 e^{t_1X_1 + t_2 X_2}]$$ and again evaluating at $t_1=t_2 = 0$ to get $E[X_1X_2]$ which we need for the covariance. Play with the derivatives and you should get all mixed moments.