Chapter 8 Special Continuous Distributions
8.1 Continuous Uniform Random Variables
\[ X \sim Unif(a,b) \]
\[ \begin{equation} f(x) = \begin{cases} \frac{1}{b-a} & \text{if } a \le x \le b\\ 0 & \text{otherwise} \end{cases} \end{equation} \]
\(X\) is uniformly distributed over the interval \([a,b]\), so at all points we have the same height for the PDF. Any two intervals of the same length can occur with equal probability.
Center and Dispersion
\[ E(X) = \frac{a+b}{2} \qquad Var(X) = \frac{(b-a)^2}{12} \qquad \sigma_x = \frac{b-a}{\sqrt{12}} \]
Additional Notes
There is also a discrete uniform random variable which is similarly defined.
A piece-wise constant pdf has a total area of 1, but can have discontinuities.
8.2 Exponential Random Variables
Models the inter-arrival time as a random variable, or amount of time until an event occurs.
\[ X \sim Exponential(\lambda) \\ \text{or} \\ X \sim Exponential(\theta = \frac{1}{\lambda}) \]
\[ \begin{equation} f_X(x) = F'(x) = \begin{cases} \lambda e^{-\lambda x} & \text{if } x \ge 0 \\ 0 & x < 0 \end{cases} \end{equation} \]
Center and Dispersion
\[ E(X) = \sigma_x = \frac{1}{\lambda} \qquad Var(X) = \frac{1}{\lambda^2} \]
Additional Notes
- Memoryless property
- Relationship between Exponential and Geometric
\[ \begin{equation} f_X(x) = F'(x) = \begin{cases} \lambda e^{-\lambda x} & \text{if } x \ge 0 \\ 0 & x < 0 \end{cases} \end{equation} \]
8.3 Gaussian Random Variables
8.3.1 Standard Normal Random Variable
A special case of the normal distribution that is used to calculated probabilities for any normal distribution, without having to calculate using it’s PDF.
\[ Z\sim N(0,1) \]
\[ f(z) = \frac{1}{\sqrt{2\pi}}\exp [-\frac{z^2}{2}] \]
\[ F(z) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^z\exp [\frac{-t^2}{2}]dt \]
Center and Dispersion
\[ E(X) = 0 \qquad Var(X) = 1 \]
Additional Notes
- We standardize