For time $t > 0$, let $X_t$ be a random variable normally distributed with
$$\mu = 0.08t$$
$$\sigma = 0.02\sqrt{t}$$
Let $S_0$ denote the known spot price of an asset today and let the random variable $S_t$ denote the unknown price of the asset in time $t$ years.
Assume $$St = S_0e^{Xt}$$
(a) If you invest all your personal money today in the asset, by when do you expect your personal worth to triple?
(b) What is the probability that your worth at least triples after two year?
For $a$ I set it up so $$E(S_0e^{\mu t +0.02\sqrt{t} Z})$$
Then I used MGF and solved for t.
Part $b$ is where I am stuck. I have it set up so $$P(S_0e^{\mu t +0.02\sqrt{t} Z}> 3S_0)$$ where t=2
From here I solve for Z but the Z value I get is really high. So I assume I made a mistake somewhere, or I have the whole thing set up wrong. Where would is this mistake?