# Probability & Random Processes - Final Exam Review

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

## Set Theory

1. $P[A\cap B] = P[AB]$
2. $P[A] \geq 0$
3. $P[\varnothing ] = 0$
4. $P[A \cup B] = P[A] + P[B] - P[A \cap B]$
5. $P[A^c] = 1 - P[A]$
6. $P[A \vert B ] = \frac{P[AB]}{P[B]}$
7. $P[B\vert B] = 1$
8. $\text{Bayes Theorem: } P[B\vert A] = \frac{P[A\vert B]P[B]}{P[A]}$
9. $\text{n choose k} = \begin{pmatrix} n\\k \end{pmatrix} = \frac{n!}{k!(n-k)!}$

## 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 $$P_X(x) = \begin{cases} 1-p , & x = 0 \\ p , & x = 1 \\ 0, & \text{otherwise} \end{cases}$$ $$p$$ $$p(1-p)$$
Geometric number of trials until 1st success $$P_X(x) = \begin{cases} p(1-p)^{x-1}, & x = 0 \\0, & \text{otherwise} \end{cases}$$ $$\frac{1}{p}$$ $$\frac{1-p}{p^2}$$
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 $$(a, b)$$ $$\begin{cases} \frac{1}{b-a} & a \leq x < b \\ 0 & \text{otherwise} \\ \end{cases}$$ $$\begin{cases} 0 & x \leq a \\ \frac{x-a}{b-a} & a < x \leq b \\ 1 & x > b \\ \end{cases}$$ $$\frac{b+a}{2}$$ $$\frac{(b-a)^2}{12}$$ Exponential $$(\lambda)$$ $$\begin{cases} \lambda e^{-\lambda x} & x \geq 0 \\ 0 & \text{otherwise} \\ \end{cases}$$ $$\begin{cases} 1- e^{-\lambda x} & x \geq 0 \\ 0 & \text{otherwise} \\ \end{cases}$$ $$\frac{1}{\lambda}$$ $$\frac{1}{\lambda^2}$$ Erlang $$(n, \lambda)$$ $$\begin{cases} \frac{\lambda x^{n-1}e^{-\lambda x}}{(n-1)!} & x > 0 \\ 0 & \text{otherwise} \\ \end{cases}$$ $$\frac{n}{\lambda}$$ $$\frac{n}{\lambda^2}$$ 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

1. $P[x_1 < x < x_2] = F_X(x_2) - F_X(x_1)$
2. $\int\limits_{-\infty}^\infty f_X(x)dx = 1$

#### Expected Value

1. $E[X] = \int\limits_{-\infty}^\infty xf_X(x) dx$
2. $E[X-\mu_x]= 0$
3. $E[aX + b] = aE[X] + b = 0$
4. $E[X+Y] = E[X] + E[Y]$

#### Variance

1. $Var[X] = E[X^2] - \mu_X^2$
2. $Var[X] = E[(X-\mu_X)^2]$
3. $Var[aX + b] = a^2Var[X]$
4. $Var[X+Y] = Var[X] + Var[Y] + 2Cov[X, Y]$

#### Covariance & Correleational Coefficients

1. $\rho_{X,Y} = \frac{Cov[X, Y]}{\sqrt{Var[X]Var[Y]}}$
2. $-1 \leq \rho_{X,Y} \leq 1$
3. $r_{X,Y} = E[XY]$
4. $Cov[X,Y] = r_{X,Y} - \mu_X\mu_Y$
5. $Cov[X,X] = Var[X] = E[X^2] - (E[X])^2$

#### Independent Variables

Given two independent random variables:

1. $f_{X,Y}(x, y) = f_X(x)f_Y(y)$
2. $r_{X,Y} = E[XY] = E[X]E[Y]$
3. $Cov[X,Y] = \rho_{X,Y} = 0$
4. $Var[X+Y] = Var[X] + Var[Y]$
Probability & Random Processes - Final Exam Review - zac blanco