# Probability & Random Processes - Final Exam Review

This page will just contain the most essential equations that pertain to topics on the final exam

## Table of Discrete Random Variables

Name of Random Variable Description of variable PMF Function Expected Value- $E[X]$ Variance - $Var[X]$
Bernoulli number successes in 1 trial $% $ $p$ $p(1-p)$
Geometric number of trials until 1st success $% $ $\frac{1}{p}$ $\frac{1-p}{p^2}$
Binomial number of successes in n trials $P_X(x) = \begin{pmatrix} n \\ x \end{pmatrix} p^x(1-p)^{n-x}$ $np$ $np(1-p)$
Pascal number of trials until k successes $P_X(x) = \begin{pmatrix} x-1 \\ k-1 \end{pmatrix} p^k(1-p)^{x-k}$ $\frac{k}{p}$ $\frac{k(1-p)}{p^2}$
Poisson Probability of x arrivals in T seconds $P_X(x) = \begin{cases} \frac{\alpha^xe^{-\alpha}}{x!}, x = 0, 1, 2\dots \\ 0,\text{otherwise} \\ \end{cases}$ $\alpha$ $\alpha$
Discrete Uniform Probability of any value between k and l $P_X(x) = \begin{cases} \frac{1}{l - k + 1},\ x=k, k+1, k+2\dots \\ 0, \text{otherwise} \end{cases}$ $\frac{k + l}{2}$ $\frac{(l-k)(l-k+2)}{12}$

## Table of Continuous Random Variables

 Name Form PDF CDF $E[X]$ $Var[X]$ Uniform $(a, b)$ $% $ $% b \\ \end{cases} %]]>$ $\frac{b+a}{2}$ $\frac{(b-a)^2}{12}$ Exponential $(\lambda)$ $% $ $% $ $\frac{1}{\lambda}$ $\frac{1}{\lambda^2}$ Erlang $(n, \lambda)$ $% 0 \\ 0 & \text{otherwise} \\ \end{cases} %]]>$ $\frac{n}{\lambda}$ $\frac{n}{\lambda^2}$ Gaussian $(\mu, \sigma)$ $\frac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/2\sigma^2}$ *$\Phi(z) = \frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^z e^{-u^2/2} du$ $\mu$ $\sigma^2$
• * Note that this is the standard normal CDF of the Gaussian random variable. To adjust for $\mu$ and $\sigma$ use $\Phi(\frac{x - \mu}{\sigma})$

## List of Essential Equations

#### Independent Variables

Given two independent random variables: