This video gives more examples of manipulations of random variables and their density and distribution functions and introduces the log-normal distribution that is sometimes used to describe the distribution of returns for some financial products.
The law of large numbers is the culmination of this course and arcs straight back to the intuitive understanding of probability as frequency of occurrence. The law states that the average $$ \frac{X_1+X_2+\dots+X_n}{n} $$ of $n$ independent random variables with common mean $\mu$ and common variance $\sigma^2$ converges in probability to $\mu$ as $n$ tends to $\infty$. The latter amounts to $$ P \Bigl( \big | \frac{X_1+X_2+\dots+X_n}{n}-\mu\big |\geq \varepsilon\Bigr) \longrightarrow 0\text{ as }n\to\infty $$ for any fixed $\varepsilon>0$. The result is obtained by means of the useful Chebyshev inequality $$ P \Bigl( \big | X-E(X)\big |\geq \varepsilon\Bigr)\leq \frac{\operatorname{var(X)}}{\varepsilon^2} $$ and requires the introduction of the concept of independence for random variables.