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ho8_1 - CS237. Problem Set 8: Variance, Geometric...

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Unformatted text preview: CS237. Problem Set 8: Variance, Geometric Distribution and Coupon Collector Problem Graded Problems due Thurs March 31 11:59PM. April 9, 2011 Reading. Schaums Chapter 5.4, 5.5. M&U 2.4 and 3.2 and Lemma 3.8 on page 51. 1 Practice Problems. Exercise 1. In class we proved that if X 1 ,X 2 are independent random variables, then VAR[ X 1 + X 2 ] = VAR[ X 1 ] + VAR[ X 2 ] (1) 1. Show that equation (1) does not hold if X 1 and X 2 are not independent. (To do this, give a specific counterexample, i.e., give example dependent RVs X 1 and X 2 that show that (1) does not hold.) Ilir notes: Consider the following example. Roll a die n times. Let X be the number of 6 and Y be the number of 1 . X i is indicator if on the i th run 6 appeared, and Y i is the indicator if on the i th trial 1 appeared. We have X = X 1 + + X n and Y = Y 1 + + Y n . Now, EX = EY = n/ 6 and VAR X = VAR Y = 5 / 36 n . However, VAR( X + Y )- VAR X- VAR Y 6 = 0 . 2. Prove the general version of this theorem. This is, for independent random variables X 1 ,X 2 ,...,X n , prove that VAR[ n X i =1 X i ] = n X i =1 VAR[ X i ] (2) Ilir notes: Lets let i = E [ X i ] . We have VAR( n X i =1 X i ) = E " n X i =1 ( X i- i ) ! 2 # = n X i =1 E ( X i- i )2 + X i 6 = j E (( X i- i )( X j- j )) = n X i =1 VAR[ X i ] + X i 6 = j E [( X i- i )( X j- j )] To complete the proof we need only show that for any independent pair X j ,X i where i 6 = j we have E [( X i- i )( X j- j )] = 0 We do this as follows: E [( X i- i )( X j- j )] = E [ X i X j- i X j- j X i + i j ] = E [ X i X j ]- i E [ X j ]- j [ X i ] + i j ] = E [ X i X j ]- i j = E [ X i ] E [ X j ]- i j = i j- i j = 0 1 were we used linearity of expectation to go from the first line to the second line, the definition of i , j to go from the second to the third line, and the independence of X i ,X j to go from the third line to the fourth. This completes our proof....
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ho8_1 - CS237. Problem Set 8: Variance, Geometric...

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