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

abondantQ 2021-10-26 Answered
Suppose that X and Y are independent rv's with moment generating functions MX(t) and MY(t), respectively. If Z=X+Y, show that MZ(t)=MX(t)MY(t).
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Expert Answer

Laith Petty
Answered 2021-10-27 Author has 103 answers
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
MZ(t)=MX(t)My(t)...(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
MZt=MXtMYt
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