# Discrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand DIS 10b

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1 CS 70 Dscrete Mathematcs ad Probablty Theory Fall 206 Sesha ad Walrad DIS 0b. Wll I Get My Package? Seaky delvery guy of some compay s out delverg packages to customers. Not oly does he had a radom package to each customer, he teds to ope a package before delverg wth probablty 2. Let X be the umber of customers who receve ther ow packages uopeed. (a) Compute the expectato E(X). (b) Compute the varace Var(X). Soluto: (a) Defe X = Hece E(X) = E( { f the -th customer gets hs/her package uopeed 0 otherwse X ) = E(X ) E(X ) = Pr[X = ] = 2 sce the -th customer wll get hs/her ow package wth probablty ad t wll be uopeed wth probablty 2 ad the delvery guy opes the packages depedetly. Hece E(X) = 2 = 2. (b) To calculate Var(X), we eed to kow E(X 2 ). By learty of expectato: E(X 2 ) = E ( (X +X X ) 2) = E( The we cosder two cases, ether = j or j. Hece E(X 2) = E(X X j ) = E(X 2 ) + E(X X j ) X X j ) =E(X X j ) 2 for all. To fd E(X X j ), we eed to calculate Pr[X X j = ]. Pr[X X j = ] = Pr[X = ]Pr[X j = X = ] = 2 2( ) sce f customer has receved hs/her ow package, customer j has choces left. Hece E(X 2 ) = 2 + ( ) 2 2( ) = 3 4. Var(X) = E(X 2 ) E(X) 2 = = Varace Ths problem wll gve you practce usg the stadard method to compute the varace of a sum of radom varables that are ot parwse depedet (so you caot use learty of varace). CS 70, Fall 206, DIS 0b

2 (a) A buldg has floors umbered,2,...,, plus a groud floor G. At the groud floor, m people get o the elevator together, ad each gets off at a uformly radom oe of the floors (depedetly of everybody else). What s the varace of the umber of floors the elevator does ot stop at? (I fact, the varace of the umber of floors the elevator does stop at must be the same, but the former s a lttle easer to compute.) (b) A group of three freds has books they would all lke to read. Each fred (depedetly of the other two) pcks a radom permutato of the books ad reads them that order, oe book per week (for cosecutve weeks). Let X be the umber of weeks whch all three freds are readg the same book. Compute Var(X). Soluto: (a) Let X be the umber of floors the elevator does ot stop at. As the prevous homework, we ca represet X as the sum of the dcator varables X,...,X, where X = f o oe gets off o floor. Thus, we have ( ) m E(X ) = Pr[X = ] =, ad from learty of expectato, E(X) = ( ) m E(X ) =. To fd the varace, we caot smply sum the varace of our dcator varables. However, we ca stll compute Var(X) = E(X 2 ) E(X) 2 drectly usg learty of expectato, but ow how ca we fd E(X 2 )? Recall that E(X 2 ) = E((X X ) 2 ) = E(X X j ) = E(X X j ) = E(X 2 ) + E(X X j ). The frst term s smple to calculate: E(X 2) = 2 Pr[X = ] = ( ) m, meag that ( ) m E(X 2 ) =. X X j = whe both X ad X j are, whch meas o oe gets off the elevator o floor ad floor j. Ths happes wth probablty ( ) 2 m Pr[X = X j = ] = Pr[X = X j = ] =. CS 70, Fall 206, DIS 0b 2

3 Thus, we ca ow compute ( ) 2 m E(X X j ) = ( ). Fally, we plug to see that ( ) m ( ) 2 m ( ( ) m ) 2 Var(X) = E(X 2 ) E(X) 2 = + ( ). (b) Let X,...,X be dcator varables such that X = f all three freds are readg the same book o week. Thus, we have ( ) 2 E(X ) = Pr[X = ] =, ad from learty of expectato, As before, we kow that E(X) = E(X 2 ) = E(X ) = ( ) 2 =. E(X 2 ) + E(X X j ). Furthermore, because X s a dcator varable, E(X 2) = 2 Pr[X = ] = ( ) 2, ad ( ) 2 E(X 2 ) = =. Aga, because X ad X j are dcator varables, we are terested Pr[X = X j = ] = Pr[X = X j = ] = (( )) 2, the probablty that all three freds pck the same book o week ad week j. Thus, ( ) E(X X j ) = ( ) (( )) 2 = ( ). Fally, we compute Var(X) = E(X 2 ) E(X) 2 = ( ) + 2 ( ). 3. Markov s Iequalty ad Chebyshev s Iequalty A radom varable X has varace Var(X) = 9 ad expectato E(X) = 2. Furthermore, the value of X s ever greater tha 0. Gve ths formato, provde ether a proof or a couterexample for the followg statemets. CS 70, Fall 206, DIS 0b 3

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