Section 8.4 Uniform and Exponential Random Variables
Random Variables whose PDFs are constant on their ranges characterize uniform random variables.
Uniform R.V.
The Uniform Distribution has 2 parameters; a and b. Both a and b are real numbers, and a<b.
A continuous r.v. X is called a Uniform R.V. if, for some finite interval (a,b) of real numbers,
its value is equally likely to lie anywhere in that interval. Equivalently its PDF is constant on that interval
and 0 elsewhere. We write X~U(a,b) when X is Uniformly distributed on the interval (a,b).
PDF:
a
b
x
f
X
−
=
1
)
(
for the interval a
≤ x ≤b, and 0 elsewhere.
CDF:
a
b
a
x
x
F
X
−
−
=
)
(
for the interval a
≤ x ≤b.
=
)
(
x
F
X
0 for x < a.
=
)
(
x
F
X
1 for x>b.
Mean is
2
b
a
+
.
Variance is
12
)
(
2
a
b
−
.
Exponential R.V.
Defn: A continuous R.V. is called an exponential R.V. if its PDF is of the form:
It can be thought of as the continuous analog of the geometric distribution.
x
X
e
x
f
λ
λ
−
=
)
(
for any x such that x is > 0
0
)
(
=
x
f
X
for x such that x is

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