Can anyone show me how to compute this MGF? I am reviewing for an exam, and the professor put an answer for the MGF of the following distribution, but did not show his work, so I have no clue how he got the answer.
$\Pr(x_1, x_2) = 6 \cdot \left(\dfrac{1}{3}\right)^{x_1} \cdot \left(\dfrac{1}{4}\right)^{x_2}$ for $1 \leq x_1,x_2 < \infty$ and $0$ elsewhere
$x_1 , x_2 \in \mathbb{Z}^{+}$
Can someone please show me how to compute the joint MGF of this distribution? Thank you for reading.
We need to evaluate$$\sum_{x_1,\,x_2\ge1}6(\tfrac13e^{t_1})^{x_1}(\tfrac14e^{t_2})^{x_2}=\frac{\tfrac12e^{t_1+t_2}}{(1-\tfrac13e^{t_1})(1-\tfrac14e^{t_2})}.$$