An analytic approach to probability is naturally established if we map the sample space to some mathematical structure suitable for analysis; this is the motivation of random variables. When extending a deterministic variable to a stochastic one, the "first order uncertainty" is variance, not expectation.
Random variable $\chi$ is a measurable mapping from a probability space to a measurable space: $\chi \in \mathcal{M}(\Omega, \Sigma; X, \Sigma_X)$ and $P(\Omega) = 1$. The codomain of a random variable is typically a Euclidean space with the Lebesgue sigma-algebra, or a Banach or Hilbert space with the Borel sigma-algebra: $(\mathbb{R}^n, \mathcal{L})$, $(H, \mathcal{B(T_d)})$. Induced sigma-algebra $\Sigma_\chi$ on the domain by a random variable is the collection of preimages of measurable sets in the codomain: $\Sigma_\chi := \{\chi^{-1}(B) : B \in \Sigma_X\}$.
Distribution $\mu: \Sigma_X \mapsto [0,1]$ of a random variable is the probability measure induced on its codomain: $\forall B \in \Sigma_X$, $\mu(B) = P(\chi^{-1}(B))$.
Cumulative distribution function (CDF) $F_\chi(x)$ of a real random variable is the real function that gives the probability measure of the half-line to the left of each value: $F_\chi(x) = \mu(-\infty, x]$. CDF is a convenient representation of the distribution: it always exists and is equivalent to the distribution if the sigma-algebra on the codomain is Borel. Probability density function (PDF) $f_\chi (x)$ of a real random variable is the derivative of a cumulative distribution function, if exists: $f_\chi (x) = \mathrm{d} F_\chi (x) / \mathrm{d} x$. Probability mass function (PMF) $f_\chi (x)$ of a discrete random variable is the real function that assigns each value of the random variable its induced measure: $f_\chi (x) = P(\chi^{-1}(x))$.
Expectation $\mathbb{E} \chi$ of a random variable is its Lebesgue integral: $\mathbb{E} \chi = \int_\Omega \chi~\mathrm{d}P$. Lebesgue integral provides a uniform definition for the expectation of discrete and continuous random variables, and ensures closure of function spaces, e.g. Banach and Hilbert spaces of functions.
Theorem (change of variables): The Lebesgue integral of a real random variable on a probability space equals the Stieltjes integral of the identity function w.r.t. the cumulative distribution function: $\int_\Omega \chi~\mathrm{d} P = \int_X x~\mathrm{d} \mu = \int_{\mathbb{R}} x~\mathrm{d} F_\chi$.
Characteristic function of a random variable can be thought of as the Fourier transform of the PDF, but unlike PDF it always exists: (1) scalar form: $\varphi_\chi (t) \equiv \mathbb{E} e^{it\chi} = \int_{\mathbb{R}} e^{itx} \mathrm{d} \mu$; (2) vector form: $\Phi_{\mathbf{\chi}}(\mathbf{w}) \equiv \mathbb{E}e^{i \mathbf{w}^T \mathbf{\chi}} = \mathcal{F} f_{\mathbf{\chi}}(\mathbf{x})$. The characteristic function uniquely determines the distribution of a random variable: $f_{\mathbf{x}}(x) = \mathcal{F}^{-1} \Phi_{\mathbf{x}}(w)$.
Weak convergence of random variables implies pointwise convergence of the corresponding characteristic functions.
If a random variable has moments up to the k-th order, then the characteristic function is $k$ times continuously differentiable on the entire real line. If a characteristic function has a k-th derivative at 0, then the random variable has moments up to the k-th order if $k$ is even, and up to the k-1-th order if $k$ is odd. The k-th moment can be computed as $\mathbb{E} \chi^K = (-i)^k \varphi_\chi^{(k)} (0)$, if the right-hand side is well defined.
Table: Standard Form of Dominant Moments
| Name | Definition | Interpretation | Dimension | Range† |
|---|---|---|---|---|
| mean | first raw moment | central tendency | as is | $(-\infty, \infty)$ |
| standard deviation | second central moment | variation | as is | $[0,\infty)$ |
| skewness | normalized third central moment | lopsidedness | dimensionless | $(-\infty, \infty)$ |
| excess kurtosis | excess normalized fourth central moment, centered at normal distribution | (for symmetric distribution) probability concentration on center and tails against the standard deviations | dimensionless | $[-2, \infty)$ |
† If exists.
Classification of positive random variables by concentration: [@Taleb2018]