Probability & Random Processes - Exam 2 Review

I’m going to use this page to help gather some of the most important information which will pertain to exam 2 and put it here!

Families 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 \((\mu, \sigma)\) \(\frac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/2\sigma^2}\) *\(\Phi(z) = \frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^z e^{-u^2/2} du\) \(\mu\) \(\sigma^2\)

List of Essential Equations

  1. \[P[x_1 < x < x_2] = F_X(x_2) - F_X(x_1)\]
  2. \[\int\limits_{-\infty}^\infty f_X(x)dx = 1\]
  3. \[E[X] = \int\limits_{-\infty}^\infty xf_X(x) dx\]
  4. \[E[X-\mu_x]= 0\]
  5. \[E[aX + b] = aE[X] + b = 0\]
  6. \[Var[X] = E[X^2] - \mu_X^2\]
  7. \[Var[X] = E[(X-\mu_X)^2]\]
  8. \[Var[aX + b] = a^2Var[X]\]
  9. \[E[X+Y] = E[X] + E[Y]\]
  10. \[Var[X+Y] = Var[X] + Var[Y] + 2Cov[X, Y]\]
  11. \[\rho_{X,Y} = \frac{Cov[X, Y]}{\sqrt{Var[X]Var[Y]}}\]
  12. \[-1 \leq \rho_{X,Y} \leq 1\]
  13. \[r_{X,Y} = E[XY]\]
  14. \[Cov[X,Y] = r_{X,Y} - \mu_X\mu_Y\]
  15. \[Cov[X,X] = Var[X] = E[X^2] - (E[X])^2\]

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 - Exam 2 Review - zac blanco