Random Variables # faulty pixels in monitor Independence # alpha - PowerPoint PPT Presentation

Random Variables Mathematics for Computer Science MIT 6.042J/18.062J Informally: an RV is a number produced by a random process: • # hours to next system crash Random Variables • # faulty pixels in monitor Independence • # alpha particles in a second • # heads in n coin flips Albert R Meyer May 6, 2013 Albert R Meyer May 6, 2013 ranvarindep.1 ranvarindep.2 Intro to Random Variables Intro to Random Variables Specify events using values of variables Example: Flip three fair coins • [C = 1] is event “exactly 1 head” C ::= # heads ( C ount) Pr[C = 1] = 3/8 • Pr[C ≥ 1] = 7/8  1 if all M atch,  =  M :: • Pr[C·M > 0] = Pr[M > 0 AND C > 0]  0 otherwis e.  = Pr[all heads] = 1/8 Albert R Meyer May 6, 2013 Albert R Meyer May 6, 2013 ranvarindep.3 ranvarindep.4 1

What is a Random Variable? What is a Random Variable? Formally, R packages together the events [R = a] for a ∈ R R: S → � Event properties carry over to RV’s directly Sample space (usually) Albert R Meyer May 6, 2013 Albert R Meyer May 6, 2013 ranvarindep.5 ranvarindep.6 Mutally Independent Variables Mutally Independent Variables Alternatively: R 1 , R 2 , … , R n Def: Pr[R 1 =a 1 AND R 2 =a 2 AND are mutually indep RV’s iff ··· AND R n =a n ] =a 2 ], …,[R n =a n ] [R 1 =a 1 ],[R 2 = Pr[R 1 =a 1 ] · Pr[R 2 =a 2 ] · are mutually indep events ··· Pr[R n =a n ] for all a 1 , a 2 , … , a n Albert R Meyer May 6, 2013 Albert R Meyer May 6, 2013 ranvar-mutual.7 ranvar-mutual.8 2

Indicator Variables Independent Variables The indicator variable for Are C and M event A:  NO  I A ::= 1 if A occurs, independent?   0 if A does not occur.  Pr[M=1] · Pr[C=1] > 0 (Sanity check: Pr[M=1 and C=1] = 0 I A and I B are independent iff A and B are independent) Albert R Meyer May 6, 2013 Albert R Meyer May 6, 2013 ranvarindep.11 ranvarindep.12 Independent Variables Independent Variables O ::= odd #Heads Lemma: Are M and I O If R is independent of S, YES then R is independent of independent? any information about S. (Work it out!) Albert R Meyer May 6, 2013 Albert R Meyer May 6, 2013 ranvarindep.13 ranvarindep.14 3

Independent Variables k-way Independent Variables k-way Independence: Lemma: any k of the variables are If R is independent of S, f : mutually independent R → R and , then R is independent of f(S) 2-way is called pairwise Albert R Meyer May 6, 2013 Albert R Meyer May 6, 2013 ranvarindep.15 ranvar-mutual.17 k-way Independent Variables Pairwise Independent Variables H i ::= indicator for Head on flip i E [1,k] Pairwise Independence sufficient k H i O :: = ⊕ i = 1 (mod 2 sum). for major applications (in later lecture). Any k of them are independent, Good to know, since pairwise holds but not k+1-way independent in important cases where mutual since any k determine the does not. remaining one. Albert R Meyer May 6, 2013 Albert R Meyer May 6, 2013 ranvar-mutual.18 ranvar-mutual.19 4

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