3.9  Uniform and Related Distributions

The distributions we consider in this section, along with a shorthand notation for each, are the:

1. uniform distribution: U(a,b),
2. multivariate uniform distribution: U_n(Ω).

3.9.1 Uniform distribution

A uniform distribution has constant probability density on an interval (a, b) and zero probability density elsewhere. The distribution is specified by two parameters: the end points a and b. We denote the distribution U(a,b). Its PDF is

[3.83]

which is illustrated in Exhibit 3.13.

Exhibit 3.13: PDF of a uniform distribution U(a,b).

A U(a,b) random variable has CDF and inverse CDF:

[3.84]

[3.85]

The expectation, standard deviation, skewness, and kurtosis of a U(a,b) random variable are:

[3.86]

[3.87]

[3.88]

[3.89]

3.9.2 Multivariate uniform distribution

Let Ω ∈ ⁿ be a bounded region with volume (area) v(Ω). The multivariate uniform distribution on Ω is denoted U_n(Ω) and has PDF

[3.90]

In applications, the distribution U_n((0,1)ⁿ) often arises. If X ~ U_n((0,1)ⁿ), its components X_i are independent random variables, each with marginal distribution U(0,1).

Exercises

3.25

If U ~ U(0,1), how is V = 1 – U distributed?
Solution

3.26

If U ~ U(0,1), how is W = bU + a distributed for any constants a, b?
Solution

3.27

Suppose U ~ U_n((0,1)ⁿ). What is Pr(U_i > 0.5 for all i)?
Solution

3.28

Suppose V ~ U_n(Ω) where the region Ω is indicated in Exhibit 3.14.

1. Is component V₁ marginally uniformly distributed?
2. Is component V₂ marginally uniformly distributed?
3. Are components V₁ and V₂ independent?

Exhibit 3.14: Region Ω for Exercise 3.28.

Solution