# Clarification of random variable with lognormal distribution (stocks)

Suppose we have a random variable $$S_t$$ with a log normal distribution distribution, where $$S_t$$ represents the price of a stock at a time $$t$$. Suppose that we have the annual volatility $$\sigma$$, of $$S_t$$. I know that the expectation of $$S_t$$ will be $$e^{\mu+1/2(\sigma)^2}$$, and that the variance will be $$(e^{(\sigma^{2})}-1)(e^{2\mu+\sigma^{2}})$$.

Questions:

1. Is the sigma here the volatility of the stock?
2. Why isn't the expectation of $$S_t$$ equal to $$\mu$$?
3. Why isn't variance equal to $$\sigma^{2}$$?
4. If we wanted to find something like $$p(40\le S_{.4}\le55)$$, how would we go about it?
5. How do we find the values $$\sigma$$ and $$\mu$$?

If $$S_t \sim LogNormal (\mu, \sigma^2)$$. It is important to remember that a LogNormal distribution $$\implies S_t > 0$$.
Now by the definition of the Log Normal distribution: $$\log S_t \sim Normal (\mu, \sigma^2)$$; $$\mu$$ is the mean of $$\log S_t$$ and $$\sigma^2$$ is the volatility (variance) of $$\log S_t$$. $$\log(x)$$ here is the natural log; $$\log (x) = log_e(x) = \ln (x)$$. Notice how $$\mu \in \mathbb{R}$$ but $$S_t > 0$$ so automatically $$\mu \neq E (X)$$.
If you want to find $$P(40 \leq S_{0.4} \leq 55)$$ then we just apply the definition of the LogNormal:
\begin{align} P(40 \leq S_{0.4} \leq 55) & = P( \log 40 \leq \log S_{0.4} \leq \log 55) \\ &=P\left(\frac{\log 40 - \mu}{\sigma} \leq \frac{\log S_{0.4} - \mu}{\sigma} \leq \frac{\log 55 - \mu}{\sigma} \right)\\ &= P\left(Z \leq \frac{\log 55 - \mu}{\sigma}\right) - P\left(Z \leq \frac{\log 40 - \mu}{\sigma}\right) \end{align}
where $$Z\sim Normal(0,1)$$. This final probabilities can be found bey looking at e.g. z tables, provided you have $$\mu$$ and $$\sigma$$.
How to estimate $$(\mu, \sigma)$$? Well depends on your approach to statistics, but wikipedia explains quite well how to find the maximum likelihood estimates, provided you have obtained some data. You could take a Bayesian approach if you prefer.