University:
University of Houston
Course:
MATH 2312 | Precalculus
Academic year:

2022
Views:

343

Pages:

9
Author:

impactwear3kz5

The Gamma Function The gamma function Γ(α) is defined by: Z ∞ Γ(α) = x α−1 e−x dx 0 45 / 53 Properties of the Gamma Function The most important properties of the gamma function are the following: 1. For any α > 1, Γ(α) = (α − 1)Γ(α − 1) 2. For any positive integer, n, Γ(n) = (n − 1)! 3. Γ( 21 ) = √ π 46 / 53 The PDF of a Gamma Distribution A continuous random variable X is said to have a gamma distribution if the pdf of X is ( 1 x α−1 e−x/β x ≥ 0 α f (x; α, β) = β Γ(α) 0 otherwise where parameters α and β satisfy α > 0, β > 0. 47 / 53 Gamma Distribution Related to the Poisson Gamma distribution is a distribution that arises naturally in processes for which the waiting times between events are relevant. It can be thought of as a waiting time between Poisson distributed events, unitl k arrivals. Thus the scale parameter can also be thought of as the inverse of the rate parameter (µ), µ1 . Then α = k and β = 1 µ In R, P(X ≤ x) = pgamma(x, α, β1 ) 48 / 53 Gamma Density Curve 1.0 Gamma Density Curve 0.0 0.2 0.4 0.6 0.8 alpha = 2, beta = 1/3 alpa = 1, beta = 1 alpha = 2, beta = 2 alpha = 2, beta = 1 0 2 4 6 8 10 49 / 53 Applications of the Gamma Distribution The gamma distribution can be used a range of disciplines including queuing models, climatology, and financial services. Examples of events that may be modeled by gamma distribution include: The amount of rainfall accumulated in a reservoir The size of loan defaults or aggregate insurance claims The flow of items through manufacturing and distribution processes The load on web servers The many and varied forms of telecom exchange 50 / 53 Example Suppose that the telephone calls arriving at a particular switchboard follow a Poisson process with an average of 5 calls coming per minute. What is the probability that up to a minute will elapse until 2 calls have come in to the switchboard? Average of 5 calls coming per minute means that β = 51 . Until 2 calls have come into the switchboard means that α = 2. 51 / 53 Mean and Variance of the Gamma Distribution The mean and variance of a random variable X having the gamma distribution are: E(X ) = µ = αβ Var (X ) = σ 2 = αβ 2 52 / 53 Example of Gamma Distribution Suppose that a transistor of a certain type is subjected to an accelerated life test, the lifetime Y (in weeks) has a gamma distribution with a mean of 24 and a standard deviation of 12. 1. Find the values of α and β. 2. Find P(Y ≤ 24) 53 / 53