1. Moment Generating for Normal

Suppose that X~N(Ux,VARx) and that Y~N(Uy,VARy).

1). Write down the moment generating function Mx(t) = E(e^tx) and My(t) = E(e^ty) of X and Y respectively.

2). If X and Y are independent random variables, derive the moment generating function of the new random variable W=1.2X + 1.5Y.

3). Identify the distribution of W, its expected value and its variance.

2. Hello,
Originally Posted by Jontan
Suppose that X~N(Ux,VARx) and that Y~N(Uy,VARy).

1). Write down the moment generating function Mx(t) = E(e^tx) and My(t) = E(e^ty) of X and Y respectively.
The mgf of a normal distribution is $M_X(t)=\exp\left(U_X t+\sigma^2_X \cdot\frac{t^2}{2}\right)$
$M_Y(t)=\exp\left(U_Y t+\sigma^2_Y \cdot\frac{t^2}{2}\right)$

2). If X and Y are independent random variables, derive the moment generating function of the new random variable W=1.2X + 1.5Y.
$M_W(t)=\mathbb{E}\left(e^{tW}\right)=\mathbb{E}\le ft(e^{1.2tX+1.5tY}\right)=\mathbb{E}\left(e^{1.2tX }\cdot e^{1.5tY}\right)$

Since X and Y are independent, the expectation of this product is the product of the expectations :

$M_W(t)=\mathbb{E}\left(e^{1.2tX}\right)\cdot\mathb b{E}\left(e^{1.5tY}\right)=M_X(1.2t)\cdot M_Y(1.5t)=\exp(\dots)$

3). Identify the distribution of W, its expected value and its variance.
Differentiate the mgf, as you must've been taught

3. For part 2, would it become: Mw(t) = (e^(1.2Ux + 1.5Uy)t + (1.44VARx + 2.25VARy)(t^2)/2 ?

For part 3: W~N(1.2Ux + 1.5Uy, 1.44VARx + 2.25VARy)

Where E(X) = 1.2Ux + 1.5Uy and Var(X) = 1.44VARx + 2.25VARy ?

Thanks alot!! xxx

4. Originally Posted by Jontan
For part 2, would it become: Mw(t) = (e^(1.2Ux + 1.5Uy)t + (1.44VARx + 2.25VARy)(t^2)/2 ?
Yes

Except that the parenthesis are not correct.

It should be $\exp\left((1.2U_X+1.5U_Y)t+(1.44\sigma^2_X+2.25\si gma^2_Y)\cdot\frac{t^2}{2}\right)$

For part 3: W~N(1.2Ux + 1.5Uy, 1.44VARx + 2.25VARy)

Where E(W) = 1.2Ux + 1.5Uy and Var(W) = 1.44VARx + 2.25VARy ?
And note that you could've got the expectation and the variance of W directly, by using these properties :
E(aX)=aE(X)
E(X+Y)=E(X)+E(Y)
Var(aX)=aČVar(X)
Var(X+Y)=Var(X)+Var(Y) if X and Y are independent.

Thanks alot!! xxx
You're welcome