# Uniform Distribution

When you ask for a random set of say 100 numbers between 1 and 10, you are looking for a sample from a continuous uniform distribution, where α = 1 and β = 10 according to the following definition.

Definition 1: The continuous uniform distribution has probability density function (pdf) given by where α and β are any parameters with α < β.

Observation: The corresponding cumulative distribution function (cdf) is The inverse cumulative distribution function is

I(p) = α + p(β − α)

Other key statistical properties are:

• Mean = (α + β) / 2
• Median = (α + β) / 2
• Mode = any xα ≤ x ≤ β
• Range = (-∞, ∞)
• Variance = (β – α)2 / 12
• Skewness = 0
• Kurtosis = -1.2

Real Statistics Functions: Excel doesn’t provide any functions for the uniform distribution. Instead you can use the following functions provided by the Real Statistics Resource Pack.

UNIFORM_DIST(x, α, β, cum) = the pdf of the continuous uniform distribution f(x) at x when cum = FALSE and the corresponding cumulative distribution function F(x) when cum = TRUE.

UNIFORM_INV(p, α, β) = x such that UNIFORM_DIST(x, α, β, TRUE) = p. Thus UNIFORM_INV is the inverse of the cumulative distribution version of UNIFORM_DIST.

Statistics with R for Business Analysts – Normal Distribution

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