# Probability/Moment Generating Function

Tags:
1. Oct 14, 2014

### tiger2030

1. The problem statement, all variables and given/known data
Let X ~ Normal(μ,σ2). Define Y=eX.
a) Find the PDF of Y.
b) Show that the moment generating function of Y doesn't exist.

2. Relevant equations

3. The attempt at a solution
For part a, I used the fact that fy(y) = |d/dy g-1(y)| fx(g-1(y)). Therefore I got that fy(y)= (1/y)(1/√(2piσ2)e-(ln(y)-μ)2/2σ2

Then for b), I used ψY(t)=E(etY)=∫etyfy(y)dy. When I plug in fy(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.

2. Oct 14, 2014

### 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.

3. Oct 14, 2014

### tiger2030

Where do I start in showing that (1/y)ety-ln(y)2/(2σ2)+2μln(y)/(2σ2) does not converge?