# 05pmf - EECS 501 PDFs AND PMFs AND MIXTURES Fall 2001 DEF A...

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EECS 501 PDFs AND PMFs AND MIXTURES Fall 2001 DEF: A discrete random variable (rv) is a mapping x : Ω countable set. DEF: A continuous random variable is a mapping x : Ω 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 , 0 < a < 1. Note: X =1 (1 - a ) X - 1 a = a/ (1 - (1 - a )) = 1 and 0 < (1 - a ) X - 1 a < 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. If 1 st coin tails, spin wheel of fortune and multiply the result by 5.

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