Download Chapter 4 - Lecture 4 The Gamma Distribution and its Relatives

Outline
Gamma Distribution
Exponential Distribution
Other Distributions
Exercises
Chapter 4 - Lecture 4
The Gamma Distribution and its Relatives
Andreas Artemiou
Novemer 2nd, 2009
Andreas Artemiou
Chapter 4 - Lecture 4 The Gamma Distribution and its Relative
Outline
Gamma Distribution
Exponential Distribution
Other Distributions
Exercises
Gamma Distribution
Gamma function
Probability distribution function
Moments and moment generating functions
Cumulative Distribution Function
Exponential Distribution
Definition
Moments, moment generating function and cumulative
distribution function
Other Distributions
Exercises
Andreas Artemiou
Chapter 4 - Lecture 4 The Gamma Distribution and its Relative
Outline
Gamma Distribution
Exponential Distribution
Other Distributions
Exercises
Gamma function
Probability distribution function
Moments and moment generating functions
Cumulative Distribution Function
Gamma Function
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In this lecture we will use a lot the gamma function.
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For α > 0 the gamma function is defined as follows:
Z ∞
Γ(α) =
x α−1 e −x dx
0
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Properties of gamma function:
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Γ(α) = (α − 1)Γ(α − 1)
For
n, Γ(n) = (n − 1)!
integer
√
1
Γ
= π
2
Andreas Artemiou
Chapter 4 - Lecture 4 The Gamma Distribution and its Relative
Outline
Gamma Distribution
Exponential Distribution
Other Distributions
Exercises
Gamma function
Probability distribution function
Moments and moment generating functions
Cumulative Distribution Function
Gamma Distribution
I
If X is a continuous random variable then is said to have a
gamma distribution if the pdf of X is:

x

−
 1
α−1
e β,x ≥ 0
f (x; α, β) = β α Γ(α) x


0,
otherwise
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If β = 1 then we have the standard gamma distribution.
Andreas Artemiou
Chapter 4 - Lecture 4 The Gamma Distribution and its Relative
Outline
Gamma Distribution
Exponential Distribution
Other Distributions
Exercises
Gamma function
Probability distribution function
Moments and moment generating functions
Cumulative Distribution Function
Mean, Variance and mgf
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Mean: E (X ) = αβ
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Variance: var(X ) = αβ 2
1
Mgf: MX (t) =
(1 − βt)α
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Andreas Artemiou
Chapter 4 - Lecture 4 The Gamma Distribution and its Relative
Outline
Gamma Distribution
Exponential Distribution
Other Distributions
Exercises
Gamma function
Probability distribution function
Moments and moment generating functions
Cumulative Distribution Function
Cumulative Distribution Function
I
When X follows the standard Gamma distribution then its cdf
is:
Z x α−1 −y
y
e
dy , x > 0
F (x; α) =
Γ(α)
0
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This is also called the incomplete gamma function
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If X ∼ Γ(α, β) then:
F (x; α, β) = P(X ≤ x) = F
Andreas Artemiou
x
;α
β
Chapter 4 - Lecture 4 The Gamma Distribution and its Relative
Outline
Gamma Distribution
Exponential Distribution
Other Distributions
Exercises
Gamma function
Probability distribution function
Moments and moment generating functions
Cumulative Distribution Function
Example 4.27 page 193
Andreas Artemiou
Chapter 4 - Lecture 4 The Gamma Distribution and its Relative
Outline
Gamma Distribution
Exponential Distribution
Other Distributions
Exercises
Definition
Moments, moment generating function and cumulative distributi
Exponential Distribution
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The exponential distributionis a special
case of Gamma. That
1
is if: X ∼ Exp(λ) ⇒ X ∼ Γ 1,
λ
If X is a continuous random variable is said to have an
exponential distribution with parameter λ > 0 if the pdf of X
is:
(
λe −λx , x > 0
f (x; λ) =
0, otherwise
Andreas Artemiou
Chapter 4 - Lecture 4 The Gamma Distribution and its Relative
Outline
Gamma Distribution
Exponential Distribution
Other Distributions
Exercises
Definition
Moments, moment generating function and cumulative distributi
Mean, Variance mgf and cdf
1
λ
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Mean: E (X ) =
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Variance: var(X ) =
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1
λ2
1
Mgf: MX (t) = 1
1− t
λ
−λx
F (x) = 1 − e
,x ≥ 0
Andreas Artemiou
Chapter 4 - Lecture 4 The Gamma Distribution and its Relative
Outline
Gamma Distribution
Exponential Distribution
Other Distributions
Exercises
Definition
Moments, moment generating function and cumulative distributi
Example 4.28 page 195
Andreas Artemiou
Chapter 4 - Lecture 4 The Gamma Distribution and its Relative
Outline
Gamma Distribution
Exponential Distribution
Other Distributions
Exercises
Other useful distributions
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Chi - square distribution
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t distribution
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F distribution
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Log - normal distribution
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Beta distribution
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Weibull distribution
Andreas Artemiou
Chapter 4 - Lecture 4 The Gamma Distribution and its Relative
Outline
Gamma Distribution
Exponential Distribution
Other Distributions
Exercises
Exercises
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Section 4.4 page 197
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Exercises 69, 70, 71, 72, 73, 74, 75, 76, 78, 80, 81
Andreas Artemiou
Chapter 4 - Lecture 4 The Gamma Distribution and its Relative