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
- \[P[A\cap B] = P[AB]\]
- \[P[A] \geq 0\]
- \[P[\varnothing ] = 0\]
- \[P[A \cup B] = P[A] + P[B] - P[A \cap B]\]
- \[P[A^c] = 1 - P[A]\]
- \[P[A \vert B ] = \frac{P[AB]}{P[B]}\]
- \[P[B\vert B] = 1\]
- \[\text{Bayes Theorem: } P[B\vert A] = \frac{P[A\vert B]P[B]}{P[A]}\]
- \[\text{n choose k} = \begin{pmatrix} n\\k \end{pmatrix} = \frac{n!}{k!(n-k)!}\]
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 | 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
- \[P[x_1 < x < x_2] = F_X(x_2) - F_X(x_1)\]
- \[\int\limits_{-\infty}^\infty f_X(x)dx = 1\]
Expected Value
- \[E[X] = \int\limits_{-\infty}^\infty xf_X(x) dx\]
- \[E[X-\mu_x]= 0\]
- \[E[aX + b] = aE[X] + b = 0\]
- \[E[X+Y] = E[X] + E[Y]\]
Variance
- \[Var[X] = E[X^2] - \mu_X^2\]
- \[Var[X] = E[(X-\mu_X)^2]\]
- \[Var[aX + b] = a^2Var[X]\]
- \[Var[X+Y] = Var[X] + Var[Y] + 2Cov[X, Y]\]
Covariance & Correleational Coefficients
- \[\rho_{X,Y} = \frac{Cov[X, Y]}{\sqrt{Var[X]Var[Y]}}\]
- \[-1 \leq \rho_{X,Y} \leq 1\]
- \[r_{X,Y} = E[XY]\]
- \[Cov[X,Y] = r_{X,Y} - \mu_X\mu_Y\]
- \[Cov[X,X] = Var[X] = E[X^2] - (E[X])^2\]
Independent Variables
Given two independent random variables:
- \[f_{X,Y}(x, y) = f_X(x)f_Y(y)\]
- \[r_{X,Y} = E[XY] = E[X]E[Y]\]
- \[Cov[X,Y] = \rho_{X,Y} = 0\]
- \[Var[X+Y] = Var[X] + Var[Y]\]