Engineering Analysis/Distributions

Distributions

There are a number of common distributions, that are used in conjunction with random variables.

Uniform Distribution

The uniform distribution is one of the easiest distributions to analyze. Also, uniform distributions of random numbers are easy to generate on computers, so they are typically used in computer software.

$f_X(x) = \left\{ \begin{matrix}\frac{1}{b - a} & \mbox{ if } a < x < b \\0 & \mbox{ otherwise}\end{matrix}\right.$

$F_X(x) = \left\{ \begin{matrix}0 & \mbox{ if } x < a \\ \frac{x}{b - a} & \mbox{ if } a < x < b \\1 & \mbox{ if } x \ge b\end{matrix}\right.$

Gaussian Distribution

The gaussian distribution, or the "normal distribution" is one of the most common random distributions. A gaussian random variable is typically called a "normal" random variable.

$f_X(x) = \mathcal{N}(\mu, \sigma^2) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{(x - \mu)^2}{\sigma^2}}$

Where μ is the mean of the function, and σ² is the variance of the function. we will discuss both these terms later.