# Thread: Moment generating functions

1. ## Moment generating functions

Independent random variables
X; Y and Z are identically distributed. Let

W
= X+Y. The moment generating function of W is MW(t) = (0.7+0.3*e^t)^6:

Find the moment generating function of V = X + Y + Z:

I have started it with
Mv(t)=E[e^(X+Y+Z)*t]
=E[(e^(X+Y)*t)*(e^Z)*t]
since X,Y,Z are independent .we can write
Mv(t)=E[e^((X+Y)*t)]*E[e^(Z*t)]
=Mw(t)*Mz(t)
Here Mw(t) is given ,but how do I find out Mz(t)..
help me to solve this

2. Originally Posted by Pushpa
Independent random variables

X; Y and Z are identically distributed. Let

W
= X+Y. The moment generating function of W is MW(t) = (0.7+0.3*e^t)^6:
Find the moment generating function of V = X + Y + Z:

I have started it with
Mv(t)=E[e^(X+Y+Z)*t]
=E[(e^(X+Y)*t)*(e^Z)*t]
since X,Y,Z are independent .we can write
Mv(t)=E[e^((X+Y)*t)]*E[e^(Z*t)]
=Mw(t)*Mz(t)
Here Mw(t) is given ,but how do I find out Mz(t)..
help me to solve this
There is a theorem that says that if $\displaystyle X_1$, $\displaystyle X_2$, .... $\displaystyle X_n$ are independent random variables with mgf's $\displaystyle m_{X_1}(t)$, $\displaystyle m_{X_2}(t)$, .... $\displaystyle m_{X_n}(t)$ then the mgf of $\displaystyle U = X_1 + X_2 + .... + X_n$ is $\displaystyle m_U(t) = m_{X_1}(t) \cdot m_{X_2}(t) \cdot .... \cdot m_{X_n}(t)$.
Using this theorem it seems to me that the since X, Y and Z are i.i.d. random variables then the mgf of V = X + Y + Z is $\displaystyle m_V = (0.7+0.3 e^t)^9$ (and that, in fact, X, Y and Z are binomial random variables with n = 3 and p = 0.3).