$\mathbb{E}[S(t)]$ and $\mathbb{E}[S(t)^2]$ of the following pdf have to be calculated:
$$f(x) = \frac{exp(\frac{-(log(x/S_0)-(\mu-\sigma^2/2)t)^2}{2\sigma^2t})}{x\sigma\sqrt{2\pi}t}$$ and S(t) being the following function:
$$S(t) = S_0e^{(\mu - \frac{1}{2}\sigma^2)t + \sigma\sqrt{t}Z}$$ with $Z\sim N(0,1)$.
Now, $\mathbb{E}[S(t)] = \int_{-\infty}^\infty Sf(S)ds$. I've tried to solve this, but I cannot seem to figure out how. I have to proof that $\mathbb{E}[S(t)] = S_oe^{\mu t}$. I've just started with mathematical finance, so I'm not all too familiar with it yet and my professor is not really being of any help. I did some research online and saw some use of Ito's lemma on functions somewhat similar to these.
Thank you in advance!
