abondantQ
2021-10-26
Answered

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

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

${M}_{Z}\left(t\right)={M}_{X}\left(t\right){M}_{y}\left(t\right)$ ...(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

${M}_{Z}t={M}_{X}t{M}_{Y}t$

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

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

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