Presentation on Probability Genrating Function
- 1. Probability Generating Function (PGF) Riaz Khan December 2, 2016 Statistical Inference II SDSU, FA 2016
- 2. The Probability Generating Function (PGF) of a discrete random variable is a power series representation of the PMF of the random variable. Mathematically, = = ∑ ( ) = ∑ ( = ) Example: ~ , = = + = = + , = − Definition 2
- 3. PGF of some common distributions 3 Distribution Probability Generating Function (PGF) Degenerate distribution, ~ ( ) = Geometric distribution, ~ ( ) ( ) = − , = − < Bernoulli distribution, ~ ( ) = + , = − Binomial distribution, ~ ( , ) = ( + ) , = − Poisson distribution, ~ ( ) = ( ) Negative binomial distribution, ~ ( , ) ( ) = − , = − <
- 4. If is a discrete RV and is the PGF of , then (a) = = (b) = Recovery of PMF from PGF: If X is a discrete RV with PGF = = = ( )| ! = ( ) ! Properties of PGF 4
- 5. Uniqueness: If two RV’s have the same PGF, then the two RV’s have the same distribution. PGF of sum of RV’s: If = + +, … + ( , , … , ) a is the PGF of , then = Properties of PGF (contn’d) 5
- 6. Linear Transformation of a RV: If = + a is the PGF of , then = ( ) Properties of PGF (contn’d) 6
- 7. Theorem: If is a discrete RV and is the PGF of , then Moments of a RV using PGF 7 First Moment: If = in ( ), then | = = − − … ( − + ) … . ( ) = ( )
- 8. Moments of RV using PGF (contn’d) 8 = ( − ) = − = − Second Moment: If = in ( ), then ∴ = + = + Similarly, third moment can be calculated as = + + Can be extended for any moment...
- 9. Moment Generating Function from PGF 9 If is a discrete RV; and and are the PGF and MGF of , respectively, then = where the MGF is defined by, =
- 10. References 10 1. https://en.wikipedia.org/wiki/Probability-generating_function 2. http://www.cl.cam.ac.uk/teaching/0708/Probabilty/prob06.pdf 3. http://www.math.uah.edu/stat/expect/Generating.html 4. http://www.am.qub.ac.uk/users/g.gribakin/sor/Chap3.pdf