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MA/STAT416: Probability
Lecture Notes
Spring 2011 1
Chapter 8: Special Continuous Distributions
April 11, 2011
8
Special Densities
8.1
The Uniform Distribution
Deﬁnition.
Let
X
be a continuous random variable, the
Uniform distribution
with
parameters
a,b
;
X
∼
U
[
a,b
], is given by the following pdf
f
(
x
) =
1
b

a
,
a
≤
x
≤
b.
Theorem.
Suppose
X
∼
U
[
a,b
]. Then,
μ
X
=
E
(
X
) =
a
+
b
2
,
and
σ
2
X
=
V ar
(
X
) =
(
b

a
)
2
12
.
Example
1
.
Let
X
∼
U
[
a,b
]. If its mean is 2 and variance is 3, what are the values of
a
and
b
?
8.2
The Exponential Distribution
Deﬁnition.
Let
X
be a continuous random variable, the
Exponential distribution
with
parameter
λ >
0;
X
∼
Exp
(
λ
), is given by the following pdf
f
(
x
) =
1
λ
e

x
λ
,
x >
0
.
Deﬁnition.
The Gamma function is deﬁned by
Γ(
α
) =
Z
∞
0
x
α

1
e

x
dx,
α >
0
.
Remark.
In particular,
Γ(
n
) = (
n

1)!
,
n
= 1
,
2
,
3
,
···
.
Γ(
α
+ 1) =
α
·
Γ(
α
)
,
∀
α >
0
.
Γ
±
1
2
²
=
√
π.
Theorem.
Suppose
X
∼
Exp
(
λ
). Then,
μ
X
=
E
(
X
) =
λ,
and
σ
2
X
=
V ar
(
X
) =
λ
2
.
Example
2
.
Suppose
X
∼
Exp
(
λ
). Answer the following:
1. Find the CDF of
X
; i.e.
F
(
x
).
2. Evaluate the
k
th
moment of
X
for a general
k
.
3. Evaluate
P
(
X > x
+
y

X > x
).
This shows that the Exponential distribution also
has the Lack of Memory property
.
Example
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This note was uploaded on 04/23/2011 for the course STAT 416 taught by Professor Staff during the Spring '08 term at Purdue University.
 Spring '08
 Staff
 Probability

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