I'm trying to prove that $E[e^{tX}]$ finite, given that $X = e^Z$ and that $Z \sim N(0,1)$ for any value of t. I'm thinking about two approaches:
- One approach is to use the integral definition of Expectation but I'm struggling to derive the integral since I don't know how to handle the pdf part.
can I say that $F_x(X) = F_z(ln(x))/x$ and then use $F_z(ln(x))$ instead ?
- Another approach is to use taylor's expansion and use dominated convergence theorem but I'm also struggling to find the dominating function of the series.
Could someone please help me, if one of the approaches is correct ? and how to proceed in it.