Uniform distribution

Assumptions

The uniform distribution describes continuous variabls which can take any value over a range defined by its minimum, \(a\), and its maximum, \(b\). All values in this range are possible, and all values are equally likely.

Definition

\[\begin{array}{ll} \text{Support:} & x \in [a,b] \\ \text{Parameter(s):} & a,\text{ the minimum }(a \in \mathbb{R}) \\ & b,\text{ the maximum }(b \in \mathbb{R}) \\ \text{PDF:} & f_X(x) = \frac{1}{b-a} \\ \text{CDF:} & F_X(x) = \frac{x-a}{b-a} \\ \text{Mean:} & \mathbb{E}[X]=\frac{a+b}{2} \\ \text{Variance:} & \mathbb{V}[X]=\frac{(b-a)^2}{12} \\ \end{array}\]

Visualizer

#| '!! shinylive warning !!': |
#|   shinylive does not work in self-contained HTML documents.
#|   Please set `embed-resources: false` in your metadata.
#| standalone: true
#| viewerHeight: 650

library(shiny)
library(bslib)

ui <- page_fluid(
      tags$head(tags$style(HTML("body {overflow-x: hidden;}"))),
  title = "Uniform distribution PDF",
  fluidRow(plotOutput("distPlot")),
  fluidRow(column(width=6,sliderInput("a", "Minimum (a)", min=-10, max=10, value=0)),
           column(width=6,sliderInput("b", "Maximum (b)", min=0.01, max=10, value=1))))

server <- function(input, output) {
  output$distPlot <- renderPlot({
    x <- seq(input$mu-3*input$sigma,input$mu+3*input$sigma,input$sigma/100)
    y <- dnorm(x,input$mu,input$sigma)
    xlims <- c(mean(c(-3,x[1])),mean(c(3,x[601])))
    ylims <- c(0,mean(c(dnorm(0),y[301])))
    plot(x=x,y=y,main=NULL,xlab='x',ylab='Density',type='l',lwd=2,
         xlim=xlims,ylim=ylims)
  })
}

shinyApp(ui = ui, server = server)

Properties

• The probability of observing values within a subrange \([x_0,x_1]\) is equal to proportion of the total range covered by the subrange:

\[\mathrm{if}\,a \le x_0 \le x_1 \le b:P(X \in [x_0,x_1]) = \frac{x_1-x_0}{b-a}\] * Uniform distributions are scale-location invariant, meaning that they can be moved around the number line and stretched (or pinched) using simple transformations without complicating any math. Suppose a uniform random variable \(X \sim \mathrm{Uniform}(a,b)\) and two constants \(c_1\) and \(c_2\). Then,

\[\mathrm{if}\; Y = c_1 + c_2X,\; \mathrm{then}\; Y \sim \mathrm{Uniform}(c_1 + c_2a, c_1 + c_2b)\]

Relations to other distributions

Every probability distribution is a transformation of the uniform distribution! This is in fact how computers generate samples of other distributions. Suppose we have a standard uniform variable \(U \sim \mathrm{Uniform}(0,1)\), and another random variable \(X\) from any other distribution, with some known CDF \(F_X(x) = P(X \le x)\) and its inverse, the quantile function \(F^{-1}_X(p)\). Then,

\[F^{-1}_X(U) \sim X\]

Let’s try it with an Exponential(3) distribution. First let’s remind ourselves of its CDF:

\[F_X(x) = 1 - e^{-3x}\]

Then we’ll find the inverse of that function (we can do this by hand since it’s a simple function):

\[F^{-1}_X(q) = \log(1-p)/3\]