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

8.3.2 Normal Random Variable

\[ X \sim N(\mu, \sigma^2) \]

\[ f(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp [-\frac{(x-\mu)^2}{2\sigma^2}] \qquad -\infty < x<\infty \]

Center and Dispersion

\[ E(X) = \mu \qquad Var(X) = \sigma^2 \]

Additional Notes