# 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 $$(\mu, \sigma)$$ $$\frac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/2\sigma^2}$$ *$$\Phi(z) = \frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^z e^{-u^2/2} du$$ $$\mu$$ $$\sigma^2$$
• * 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. $\int\limits_{-\infty}^\infty f_X(x)dx = 1$
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