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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Statistics and Probability
College Statistics
Random variables

abondantQ

Answered question

2021-10-26

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

M_{X} (t)

and

M_{Y} (t)

, respectively. If Z=X+Y, show that

M_{Z} (t) = M_{X} (t) M_{Y} (t)

Answer & Explanation

Laith Petty

Skilled2021-10-27Added 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

M_{Z} (t) = M_{X} (t) M_{y} (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

M_{Z} t = M_{X} t M_{Y} t

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