Applied Engineering Stochastic Processes Cheat Sheet

This cheat sheet covers fundamental concepts in stochastic processes, including multiple random variables, joint distributions, conditional probabilities, and estimation techniques, crucial for analyzing random phenomena in engineering.

Core Principles

Stochastic processes involve systems that evolve randomly over time, requiring probabilistic models for analysis.
Multiple random variables can be analyzed using joint probability distributions (PMF/PDF) and their properties.
Conditional probability and distributions are essential for updating beliefs based on new information.
Independence between random variables simplifies analysis, allowing for factorization of joint distributions.
Estimation techniques, like MMSE and LMSE, aim to find the best prediction of a random variable based on available data.
Covariance and correlation quantify the linear relationship between random variables, indicating their tendency to vary together.

Formulas

$P(A|B) = P(A \cap B) / P(B)$
$f_{X|B}(x) = f_X(x) / P[B]$ (for continuous variables)
$P_{X|B}(x) = P_X(x) / P[B]$ (for discrete variables)
$Cov[X, Y] = E[(X - \mu_X)(Y - \mu_Y)]$
$\rho_{X,Y} = Cov[X, Y] / (\sigma_X \sigma_Y)$
$E[X+Y] = E[X] + E[Y]$
$Var[X+Y] = Var[X] + Var[Y] + 2Cov[X, Y]$
$ \hat{X}_L(Y) = \rho_{X,Y} \frac{\sigma_X}{\sigma_Y}(Y - \mu_Y) + \mu_X $ (Linear MMSE Estimator)