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Unformatted text preview: EECS 501 PDFs AND PMFs AND MIXTURES Fall 2001 DEF: A discrete random variable (rv) is a mapping x : countable set. DEF: A continuous random variable is a mapping x : uncountable set. WLOG: Countable range of discrete rv is 1-1 with Z =integers. DEF: The pmf (probability mass function) for a discrete rv x is Props: p x ( X ) = Pr [ x = X ]. We have: 0 p x ( X ) 1; X =- p x ( X ) = 1. For continuous rvs, use pdf (probability density function) f x ( X ). EX1: Discrete rv x=#heads in 3 independent flips of a fair coin. p x (0) = p x (3) = 1 8 ; p x (1) = p x (2) = 3 8 ; p x ( X ) = 0 otherwise. EX2: Geometric pmf: p x ( X ) = (1- a ) X- 1 a, X 1 , &lt; a &lt; 1. Note: X =1 (1- a ) X- 1 a = a/ (1- (1- a )) = 1 and 0 &lt; (1- a ) X- 1 a &lt; 1. Note: PDF (distribution) F x ( X ) for discrete rv x is piecewise constant (staircase) function with a countable number of discontinuities. But: Flip a fair coin. If heads, flip a 2 nd coin with Pr[heads]=2/3. If 2 nd coin heads, set rv x=3; if 2 nd coin tails, set rv x=1....
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This note was uploaded on 07/22/2011 for the course EECS 370 taught by Professor Lehman during the Spring '10 term at University of Florida.
- Spring '10