Lecture8standard - Statistics 511 Statistical Methods Dr...

Statistics 511: Statistical Methods
Dr. Levine
Purdue University
Fall 2011
Lecture 10: Other Continuous Distributions and Probability
Plots
Devore: Section 4.4-4.6
October, 2011
Page 1

Statistics 511: Statistical Methods
Dr. Levine
Purdue University
Fall 2011
Gamma Distribution
•
Gamma function is a natural extension of the factorial
•
For any
α > 0
,
Γ( α ) =
Z
∞
0
x
α - 1
e
- x
dx
•
Properties:
1. If
α > 1
,
Γ( α ) = ( α - 1)Γ( α - 1)
2.
Γ( n ) = ( n - 1)!
for any
n ∈ Z
+
3.
Γ
(
1
2
)
=
√
π
October, 2011
Page 2

Statistics 511: Statistical Methods
Dr. Levine
Purdue University
Fall 2011
•
The natural definition of a density based on the gamma function
is
f ( x ; α ) =



x
α - 1
e
- x
Γ( α )
if
x ≥ 0
0
otherwise
•
A gamma density with parameters
α > 0
,
β > 0
is
f ( x ; α, β ) =



1
β α Γ( α )
x α - 1 e - x/β
if
x ≥ 0
0
otherwise
• α
is a shape parameter ,
β
is a scale parameter
•
The case
β = 1
is called the standard gamma distribution
October, 2011
Page 3

Statistics 511: Statistical Methods
Dr. Levine
Purdue University
Fall 2011
Gamma pdf: graphical illustration
October, 2011
Page 4

Statistics 511: Statistical Methods
Dr. Levine
Purdue University
Fall 2011
Gamma Distribution Parameters
•
The mean and variance of a random variable
X
having the
gamma distribution
f ( x ; α, β )
are
E ( X ) = αβ
and
V ( X ) = αβ
2
•
Let
X
have a gamma distribution with parameters
α
and
β
•
Then
P ( X ≤ x ) = F ( x ; α, β ) = F ( x/β ; α )
•
In the above,
F ( x ; α ) =
Z
x
0
y
α - 1
e
- y
Γ( α )
dy
is an incomplete gamma function . It is defined for any
x > 0
.
October, 2011
Page 5

Statistics 511: Statistical Methods
Dr. Levine
Purdue University
Fall 2011
Exponential Distribution as a Special Case of Gamma Distribution
•
Assume that
α = 1
and
β = 1
λ
.
•
Then,
f ( x ; λ ) =



λe - λx
if
x ≥ 0
0
otherwise
•
Its mean and variance are
E ( X ) =
1
λ
and
V ( X ) =
1
λ
2
•
Note that
μ = σ
in this case
October, 2011
Page 6

Statistics 511: Statistical Methods
Dr. Levine
Purdue University
Fall 2011
Exponential pdf: graphical illustration
October, 2011
Page 7

Statistics 511: Statistical Methods
Dr. Levine
Purdue University
Fall 2011
Exponential cdf
•
Exponential cdf can be easily obtained by integrating pdf, unlike
the cdf of the general Gamma distribution
•
The result is
F ( x ; λ ) =



1 - e - λx
if
x ≥ 0
0
otherwise
October, 2011
Page 8

Statistics 511: Statistical Methods
Dr. Levine
Purdue University
Fall 2011
Example I
•
Suppose the response time
X
at an on-line computer terminal
has an exponential distribution with expected response time
5
sec
• E ( X ) =
1
λ
= 5
and
λ = 0 . 2
•
The probability that the response time is between
5
sec and
10
sec is
P (5 ≤ X ≤ 10) = F (10; 0 . 2) - F (5; 0 . 2) = 0 . 233
•
In R: pexp(10,0.2)-pexp(5,0.2)
October, 2011
Page 9

Statistics 511: Statistical Methods
Dr. Levine
Purdue University
Fall 2011
Example II
•
On average,
3