Probability/Moment Generating Function

[tiger2030]

Homework Statement

Let X ~ Normal(μ,σ²). Define Y=e^X.
a) Find the PDF of Y.
b) Show that the moment generating function of Y doesn't exist.

Homework Equations

The Attempt at a Solution

For part a, I used the fact that f_y(y) = |d/dy g^-1(y)| f_x(g^-1(y)). Therefore I got that f_y(y)= (1/y)(1/√(2piσ²)e^{-(ln(y)-μ)2/2σ2}

Then for b), I used ψ_Y(t)=E(e^tY)=∫e^tyf_y(y)dy. When I plug in f_y(y) I get a function that is nonlinear and too complicated to integrate. If someone could give me a hint on the next step it would be greatly appreciated.

 

[Ray Vickson]

You need to analyze the behavior of $e^{ty} f(y)$ for $y \to + \infty$ in order to show that the integral does not converge. You do not need to actually compute the integral to do that.

 

[tiger2030]

Where do I start in showing that (1/y)e^{ty-ln(y)2/(2σ2)+2μln(y)/(2σ2)} does not converge?