Stability of the gaussian under addition and scalar multiplication

[stukbv]

If i have the mgf of X and the mgf of Y where X~N(mx,vx) and Y~N(my,vy) and X and Y are independent ,
then if i want to show that aX +bY ~ N(amx+bmy , a^2vx+b^2vy) how would i do this - need to be able to do the convolutions way and the mgf's way,
for the mgfs way is it just, mgf(ax+by) = mgf(ax) . mgf(bx) if so how do you find mgf(ax) ?

Thanks

 

[micromass]

Hi stukbv!

Sadly, there is no way to easily calculate the mgf of aX when just knowing X (or at least, I don't know such a way). The easiest is by showing in general that the mgf of $X\sim N(\mu,\sigma^2)$ is

$$M(t)=e^{t\mu+\frac{1}{2}\sigma^2t^2}$$

Then the distribution of aX is

$$M(at)=e^{at\mu+\frac{1}{2}\sigma^2a^2t^2}$$

And what kind of distribution is this?