Moment Generating Function Factorization Method

Method

For given MGF of RV $X$, The RV $Y,Z$ where $X=Y+Z$ can be found by factoring the MGF

Example

$X$ with MGF $e^{({\lambda }_1+{\lambda }_2)(e^t-1)}$ Then, $Y$ with MGF $e^{({\lambda }_1)(e^t-1)}$ Then, $Z$ with MGF $e^{({\lambda }_2)(e^t-1)}$