The Uniform Distribution
The uniform probability density function f(x) and the associated distribution
function F(x)are given by
f(x) =
{ 1
b−a, a < x < b
0 , otherwise
F(x) =
∫x
−∞
f(x)dx =
∫x
a
f(x)dx
It is sometimes referred to as the rectangular distribution on the interval a < x < b.
This distribution has the mean
μ=
∫∞
−∞
xf (x)dx =
∫b
a
x
1
b−adx =
1
2 (a+b)
and variance
σ^2 =
∫∞
−∞
(x−μ)^2 f(x)dx =^1
12
(b−a)^2
The cumulative distribution function is given by F(x) =
0 , x < a
x−a
b−a, a ≤x≤b
1 , x > b
The uniform probability density function is used in pseudo-random number genera-
tors with sampling is over the interval 0 ≤x≤ 1.
Confidence Intervals
Sampling theory is a study of the various relationships that exist between prop-
erties of a population and information obtained based upon samples from the popu-
lation. For example, each sample collected from a population has associated with it
a sample mean x=μxand sample variance s^2 =σ^2 x. How do these quantities compare
with the true population mean μand true population variance σ^2? It would be nice
to put limits, like numbers γ 1 , γ 2 , associated with the value μxso that one can write
a statement like
μx−γ 1 < μ < μ x+γ 2
