L17_CouponCollector

# L17_CouponCollector - 1-1COMP170Discrete Mathematical...

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Unformatted text preview: 1-1COMP170Discrete Mathematical Toolsfor Computer ScienceMore on“time until first success”Version 2.0: Last updated, May 13, 20072-1Example 12-2Example 1Throw a fair die until you see a1.2-3Example 1Throw a fair die until you see a1.Then throw it until you see a2.2-4Example 1Throw a fair die until you see a1.Then throw it until you see a2.Continue until you see all of3,4,5,6, in that order.2-5Example 1Throw a fair die until you see a1.How many times, on average, do you throw the die?Then throw it until you see a2.Continue until you see all of3,4,5,6, in that order.2-6Example 1Throw a fair die until you see a1.SetX1= # of throws until you see1.Fori >1 :Xi= # of throws, starting from when you seei-1for the first time, until you seeifor the first time.How many times, on average, do you throw the die?Then throw it until you see a2.Continue until you see all of3,4,5,6, in that order.2-7Example 1Throw a fair die until you see a1.SetX1= # of throws until you see1.Fori >1 :Xi= # of throws, starting from when you seei-1for the first time, until you seeifor the first time.How many times, on average, do you throw the die?Then throw it until you see a2.Continue until you see all of3,4,5,6, in that order.2 51|{z}3 1 42|{z}6 5 4 43|{z}1 34|{z}6 4 1 25|{z}1 4 36|{z}X1= 3X2= 4X3= 5X4= 3X5= 5X6= 43-1X=X1+X2+...+X6.Total number of throws is2 51|{z}3 1 42|{z}6 5 4 43|{z}1 34|{z}6 4 1 25|{z}1 4 36|{z}X1= 3X2= 4X3= 5X4= 3X5= 5X6= 43-2X=X1+X2+...+X6.Total number of throws is2 51|{z}3 1 42|{z}6 5 4 43|{z}1 34|{z}6 4 1 25|{z}1 4 36|{z}X1= 3X2= 4X3= 5X4= 3X5= 5X6= 4X= 193-3X=X1+X2+...+X6.Total number of throws isXiis a geometric random variable withp= 1/6,⇒E(Xi) =1p= 6.2 51|{z}3 1 42|{z}6 5 4 43|{z}1 34|{z}6 4 1 25|{z}1 4 36|{z}X1= 3X2= 4X3= 5X4= 3X5= 5X6= 4X= 193-4X=X1+X2+...+X6.Total number of throws isE(X) =E(X1) +E(X2) +...+E(X6) = 6·6 = 36....
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L17_CouponCollector - 1-1COMP170Discrete Mathematical...

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