# Suppose that X and Y are independent rv's with moment generating functions M

Suppose that X and Y are independent rv's with moment generating functions ${M}_{X}\left(t\right)$ and ${M}_{Y}\left(t\right)$, respectively. If Z=X+Y, show that ${M}_{Z}\left(t\right)={M}_{X}\left(t\right){M}_{Y}\left(t\right)$.
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Laith Petty
Step 1
If X and Y are independent random variable and Z = X + Y
then m.g.f of Z is equal to the product of m.g.f of X and Y
i.e
${M}_{Z}\left(t\right)={M}_{X}\left(t\right){M}_{y}\left(t\right)$...(1)
Step 2
As m.g.f is defined as the exponent of argument t and the random variable. As proposition
is used i.e product of functions of independent random variables.
Eetx Eety = Eetx ety =Eetx+y...(2)
Step 3
In expression (2)
On left hand side indicate moment generating function of z variable and on right hand side
indicate moment generating functions
Thus , statement has been proved
${M}_{Z}t={M}_{X}t{M}_{Y}t$