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L17_CouponCollector_print

# L17_CouponCollector_print - COMP170 Discrete Mathematical...

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1 COMP170 Discrete Mathematical Tools for Computer Science More on “time until first success” Version 2.0: Last updated, May 13, 2007

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2 Example 1 Throw a fair die until you see a 1 . Set X 1 = # of throws until you see 1 . For i > 1 : X i = # of throws, starting from when you see i - 1 for the first time, until you see i for the first time. How many times, on average, do you throw the die? Then throw it until you see a 2 . Continue until you see all of 3 , 4 , 5 , 6 , in that order. 2 5 1 3 1 4 2 6 5 4 4 3 1 3 4 6 4 1 2 5 1 4 3 6 X 1 = 3 X 2 = 4 X 3 = 5 X 4 = 3 X 5 = 5 X 6 = 4
3 X = X 1 + X 2 + . . . + X 6 . Total number of throws is E ( X ) = E ( X 1 ) + E ( X 2 ) + . . . + E ( X 6 ) = 6 · 6 = 36 . X i is a geometric random variable with p = 1 / 6 , E ( X i ) = 1 p = 6 . 2 5 1 3 1 4 2 6 5 4 4 3 1 3 4 6 4 1 2 5 1 4 3 6 X 1 = 3 X 2 = 4 X 3 = 5 X 4 = 3 X 5 = 5 X 6 = 4 X = 19 Then, by linearity of expectation,

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4 Example 2 Throw a fair die until you have seen all 6 numbers. Let X 1 = 1 : For i > 1 : X i = # of throws needed to get N i after first time we see N i - 1 .
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L17_CouponCollector_print - COMP170 Discrete Mathematical...

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