How to find the Moment Generating function of a function of random variables from their joint Moment Generating function?

Question

Given the Joint Moment Generating function(MGF) of the random vector $X =(Y,Z)$

$$M_{Y,Z}(t_1,t_2) =e^{(t_1^2 + t_2^2 + t_1 t_2)/2} $$

How can I find the MGF of

1. $Y+Z$
2. $Y-Z$

Is there any generalisation for the MGF of the function G(Y,Z) ?

The way by textbook has solved this is:

1. The MGF of $Y+Z$ can be obtained by putting $t_1 = t_2 = t $ in the MGF $M_{Y,Z}(t_1,t_2)$
2. The MGF of $Y-Z$ can be obtained by putting $t_1 =t $ and $ t_2 = -t $ in the MGF $M_{Y,Z}(t_1,t_2)$

I have no idea from why this is true. I think I am missing an important result. Please help me

Answer 1

Recall the definition $M_{Y,Z}(t_1, t_2):= E[e^{t_1 Y + t_2 Z}]$. Then,

$$M_{Y+Z}(t) = E[e^{t(Y+Z)}] = E[e^{tY + tZ}] = M_{Y,Z}(t, t)$$

$$M_{Y-Z}(t) = E[e^{t(Y-Z)}] = E[e^{tY + (-t)Z}] = M_{Y,Z}(t, -t)$$