Let X and Y be two independent random variables with moment

Let X and Y be two independent random variables with moment generating functions:

M_x(t) = e^{2t^2+3t} , M_Y(t) = e^{3t^2+t}

^{Determine the moment generating function of 2X+3Y}

Solution

Here we will use definition of moment generating function.

Let z=2x+3y. So M_z(t)=E(e^tz)=E(e^(2xt+3yt))=E(e^2xte^3yt)=E(e^2xt).E(e^3yt).

Now Mx(t)=e2t^2+3t and My(t)=e3t^2+t. Putting this value we will get Mz(t)= e2(2t)^2+3(2t)*e3(3t)^2+3t=(e8t^2+6t)*(e27t^2+3t).