In this video the important representation formula $$ E(Y)=\int _0^\infty P(Y>y)\, dy-\int _{-\infty}^0 P(Y<-y)\, dy $$ is derived connecting the expected value of a continuous random variable $Y$ to its cumulative distribution function. Making use of it, the proof of $$ E \bigl( g(X)\bigr)=\int _{-\infty} ^\infty g(x)\, f_X(x)\, dx $$ is also presented for a continuous random variable $X$ and a (continuous) function $g:\mathbb{R}\to \mathbb{R}$. Do you remember how the corresponding formula works for a discrete random variable?
The uniform and the normal distribution are introduced. The latter is ubiquitous in probability and applications in many sciences. It is characterized by its cumulative distribution function $$ N(\mu,\sigma^2)=\frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2 \sigma^2}} $$ where $\mu$ is its mean and $\sigma ^2$ its variance. The video ends with a couple of examples.
The De Moivre-Laplace formula connects the binomial distribution to the normal distribution through a limiting process. It is a special case of the very important central limit theorem, where similar convergence to a normal distribution is obtained for a more general sum of independent random variables. In the video a binomially distributed random variable $S_n\sim B(n,p)$ is used. It can be obtained as the sum of $n$ independent Bernoulli random variables $X_k\sim B(1,p)$, $k=1,\dots, n$. The exponential distribution is introduced next. Its properties are studied, in particular its memoryless nature, and examples are given. It is characterized by the probability density function $$ N(\lambda)(x)=\begin{cases} 0,&x<0,\\ \lambda e^{-\lambda x},&x\geq 0.\end{cases} $$