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 $\chi: \Omega \mapsto X$ is a measurable mapping on a probability space $(\Omega, \Sigma, P)$. The sigma-algebra on the codomain $X$ is denoted as $\Sigma_X$. The codomain of a random variable is typically the real line $\mathbb{R}$ with the Borel sigma-algebra $\mathcal{B(T_d)}$ or the Lebesgue sigma-algebra $\mathcal{L}$, or a Banach or Hilbert space with a sigma-algebra. Sigma-algebra introduced by a random variable is the class of preimages of measurable sets in the codomain: $\Sigma_\chi := \{\chi^{-1}(B) : B \in \Sigma_X\}$.
Distribution of a random variable $\chi: \Omega \mapsto X$ is the probability measure $\mu: \Sigma_X \mapsto [0,1]$ induced on its codomain: $\mu(B) := P(\chi^{-1}(B))$.
Cumulative distribution function (CDF) of a real random variable $\chi: \Omega \mapsto \mathbb{R}$ is the real function $F_\chi(x) := \mu(-\infty, x]$. Cumulative distribution function $F_\chi$ is a convenient characterization of the distribution $\mu$: CDF always exists, and is equivalent to the distribution if the sigma-algebra of the real line is Borel. Probability density function (PDF) of a real random variable is the derivative of a cumulative distribution function, if the derivative exists: $f_\chi (x) := \mathrm{d} F_\chi (x) / \mathrm{d} x$. Probability mass function (PMF) 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 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: $\int_\Omega \chi~\mathrm{d} P = \int_X x~\mathrm{d} \mu = \int_{\mathbb{R}} x~\mathrm{d} F_\chi$.
The characteristic function of a random variable $\chi$ with measure $\mu$ is
$$\begin{aligned} \varphi_\chi (t) &\equiv \mathbb{E} e^{it\chi} = \int_{\mathbb{R}} e^{itx} \mathrm{d} \mu && \text{(scalar form)} \\ \Phi_{\mathbf{\chi}}(\mathbf{w}) &\equiv \mathbb{E}e^{i \mathbf{w}^T \mathbf{\chi}} = \mathcal{F} f_{\mathbf{\chi}}(\mathbf{x}) && \text{(vector form)} \end{aligned}$$
The characteristic function can be thought of as the Fourier transform of the PDF. But unlike PDF, the characteristic function of a distribution always exist.
The characteristic function uniquely determines the distribution of a random variable: $f_{\mathbf{\chi}}(\mathbf(x)) = \mathcal{F}^{-1} \Phi_{\mathbf{\chi}}(\mathbf(w))$.
Weak convergence of random variables implies pointwise convergence of the corresponding characteristic functions.
If a random variable $\chi$ 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.
If the right-hand side is well defined, the $k$-th moment can be computed as:
$$\mathbb{E} \chi^K = (-i)^k \varphi_\chi^{(k)} (0)$$
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]