Uniformly Distributed Random Variables

Uniformly Distributed Random Variables

Definition.
Suppose that X is a continuous random variable assuming all
values in the interval [a, b ], where both a and b are finite. If the pdf of X is
given by
f(x)= 1/(b-a),    a ≤ x ≤ b
= 0,                   elsewhere,
(4.13)
we say that X is uniformly distributed over the interval [a, b].
Notes: 
(a) A uniformly distributed random variable has a pdf which is constant over
the interval of definition. In order to satisfy the condition \[\int_{-\infty }^{+\infty }{f\left( x \right)}dx=1\]

 this constant must be equal to the reciprocal of the length of the interval.
(b) A uniformly distributed random variable represents the continuous analog to
equally likely outcomes in the following sense. For any sub interval [c, d], where
a ≤ c < d ≤ b, P(c ≤ X ≤ d) is the same for all sub intervals having the same length.
That is,
\[P\left( c\le X\le d \right)=\int\limits_{c}^{d}{f\left( x \right)}dx=\frac{d-c}{b-a}\]
and thus depends only on the length of the interval and not on the location of that
interval.
(c) We can now make precise the intuitive notion of choosing a point P at random
on an interval, say [a, b ]. By this we shall simply mean that the x-coordinate of the chosen
point, say X, is uniformly distributed over [a, b].

EXAMPLE 1
 A point is chosen at random on the line segment [0, 2]. What
is the probability that the chosen point lies between 1 and -! ?
Letting X represent the coordinate of the chosen point, we have that the pdf of
X is given by f(x) =1/2, 0 < x < 2, and hence P(l ≤ X ≤ 3/2) = 1/4

EXAMPLE 2
 The hardness, say H, of a specimen of steel (measured on the
Rockwell scale) may be assumed to be a continuous random variable uniformly
distributed over [50, 70] on the B scale. Hence
f(h) =1/20                50 < h < 70,
       =0,                elsewhere.
EXAMPLE 3
 Let us obtain an expression for the cdf of a uniformly distributed
random variable.
F(x) = P(X ≤  x) =
\[\int_{-\infty }^{x}{f\left( s \right)}ds\]
=0                  if               x (x-a)/(b-a)     if              a ≤  x < b,
=1                                  if x ≥ b.