Uniform distribution

The uniform distribution is a continuous Probability distribution with constant Probability over an interval. For an interval (a,b)(a,b), the Probability density function of a uniform Random variable XX is

fX(x)={cif x∈(a,b)0if x∉(a,b)f_{X}(x)=\begin{cases} c & \text{if }x\in (a,b) \\ 0 & \text{if }x\notin(a,b) \end{cases}

It is also possible to define a discrete uniform distribution over nn possible outcomes as a simple probability mass function:

pX(x)=1np_{X}(x)=\frac{1}{n}

The discrete version is a simple mathematical description of a set of nn possibilities that are all equally likely to happen. For example, a fair coin toss is a discrete uniform distribution with n=2n=2, while a fair nn-sided die is, unsurprisingly, the same but with nn outcomes.

The continuous version is often used as a placeholder distribution when lacking information. For instance, if the main error on a measurement is due to tool precision constraints (absolute error ΔX\Delta X), a uniform distribution is used to model what possible values the measured variable could take in the 2ΔX2\Delta X interval.

Moments

The raw and central Moment-generating function are

MX∗(t)=1t(b−a)(ebt−eat),MX(t)=1t(b−a)(e(b−a)t/2−e−(b−a)t/2)M^{*}_{X}(t)=\frac{1}{t(b-a)}(e^{bt}-e^{at}),\qquad M_{X}(t)=\frac{1}{t(b-a)}(e^{(b-a)t/2}-e^{-(b-a)t/2})

The moments are:

  • Raw 0. μ0∗=1\mu_{0}^{*}=1
    1. μ1∗=a+b2\mu_{1}^{*}=\frac{a+b}{2} (mean)
  • Central 0. μ0=1\mu_{0}=1
    1. μ1=0\mu_{1}=0
    2. μ2=(b−a)212\mu_{2}=\frac{(b-a)^{2}}{12} (Variance)
    3. μ3=0\mu_{3}=0
    4. μ4=(b−a)480\mu_{4}=\frac{(b-a)^{4}}{80}
  • Coefficients 0. γ1=0\gamma_{1}=0 (skewness, it is symmetrical around the mean)
    1. γ2=−6/5\gamma_{2}=- 6/5 (kurtosis)

Properties

It is normalized by c=1/(b−a)c=1/(b-a).