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Variance equals the variance of a conditional mean plus the mean conditional variance.

A renewal process with independent identically distributed nonnegative interarrival times ${\xi }_1,{\xi }_2,\dots$ has renewal epochsand counting process $N_t=\max \{n:S_n\le t\}$.

A Markov chain has a future conditional distribution depending on the present state alone.

In a fair random root-to-leaf path, a specified leaf at depth $d$ is reached with probability $2^{-d}$.

A random partition is a probability distribution on partitions of a finite or countable set.

Conditional probability is $P(A\mid B)=P(A\cap B)/P(B)$ when $P(B)>0$.

Probability inequalities bound event probabilities using moments or other tractable quantities.

The expected value is the probability-weighted average of a random variable, defined by a sum or an integral when it exists.

A probability distribution assigns probabilities to the possible values of a random variable.

A Poisson point process has independent counts on disjoint sets and Poisson count in each set with mean given by its intensity measure.

A Fokker-Planck equation evolves a density by drift and diffusion in conservation-law form.

A rate-$\lambda$ Poisson process has independent stationary increments. Conditional on $N_t=n$, arrival times are the order statistics of $n$ independent uniform points on $[0,t]$.

Pointwise convergence of characteristic functions to a function continuous at zero implies convergence in distribution.

A stochastic process is a family of random variables indexed by time or another ordered parameter.

If $\sum \mathbb{P}(A_n)<\infty$, only finitely many $A_n$ occur almost surely. For independent events, divergence implies infinitely many occur. Use a union bound for the first and $\prod (1-p_n)\le e^{-\sum p_n}$ for the second.

An integrable adapted process $(M_n)$ is a martingale when $\mathbb{E}[M_{n+1}\mid {\mathcal{F}}_n]=M_n$.

A simple symmetric random walk starting from $S_0$ haswhere the increments ${\xi }_k$ are independent and satisfy

The main modes include almost-sure convergence, convergence in probability, convergence in $L^p$, and convergence in distribution.