Statistics 270 - Lecture 16. Gamma Distribution Normal pdf is symmetric and bell-shaped Not all distributions have these properties Some pdf’s give a.

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Presentation transcript:

Statistics Lecture 16

Gamma Distribution Normal pdf is symmetric and bell-shaped Not all distributions have these properties Some pdf’s give a skewed distribution One such family is the gamma family

Gamma Function For  >0, the gamma function is Properties: For any  >1,  (  )=(  -1)  (  -1) For any positive integer, n,  (n)=(n-1)!  ( 1/2)=

Gamma PDF A continuous random variable, X, has a gamma distribution if it has pdf: Where,  >0 and  >0

Gamma PDF Expected Value: Variance:

Standard Gamma PDF’s Standard gamma=incomplete gamma  = 1 Table A.4 gives tabled probabilities for incomplete gamma pdf’s for some  ’s

Example: (Chapter 4, #57) Suppose the time spent by a randomly selected student who uses a terminal connected to a local computer facility has a mean of 20 minutes and variance 80 minutes 2 Find  and  ? What is the probability that the student uses the terminal less than 24 minutes?

Example: (Chapter 4, #57) Suppose that X has a gamma(4.5,1.5) distribution Find

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Exponential Distribution

Memoryless Property