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


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



Gamma function Probability distribution function Moments and moment generating functions Cumulative Distribution Function

Gamma Function I

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: I I I

Γ(α) = (α − 1)Γ(α − 1) For n, Γ(n) = (n − 1)! integer √ 1 Γ = π 2

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Gamma Distribution

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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.

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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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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

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x ;α β




Example 4.27 page 193

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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

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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

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Example 4.28 page 195

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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

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Exercises

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Section 4.4 page 197 I

Exercises 69, 70, 71, 72, 73, 74, 75, 76, 78, 80, 81

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