# For some set a and its complement a c in the fourth

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for some set A and its complement A C in the fourth equality. Furthermore, there is only one pair of ( X 1 ; X 2 ) whose sum is greater than 1.9734, i.e. only when X 1 = X 2 = 1 : Lastly, since X 1 and X 2 are independent, P ( X 1 = 1 ; X 2 = 1) = P ( X 1 = 1) ³ P ( X 2 = 1) = 1 3 ³ 1 3 = 1 9 : (c) Similarily, when n = 3 ; P ( Z 3 µ 0 : 924) = P ° 1 p 3 3 P i =1 ( X i ´ ° ) µ 0 : 924 ± = P ° 1 p 3 ° X 1 ´ 1 3 + X 2 ´ 1 3 + X 3 ´ 1 3 ± µ 0 : 924 ± = P ( X 1 + X 2 + X 3 µ 2 : 6004) = 1 ´ P ( X 1 + X 2 + X 3 > 2 : 6004) = 1 ´ P ( X 1 = 1 ; X 2 = 1 ; X 3 = 1) = 26 27 · 0 : 963 (d) So far, we have calculated the probability of P ( Z n µ 0 : 924) for n = 2 in (b) and n = 3 in (c). Now, we calculate the probability of P ( Z n µ 0 : 924) for large n using the central limit 5

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theorem. As n ! 1 ; the central limit theorem states, p n ± ² X n ´ ° ³ ! d N (0 ; 1) where X n = 1 n P n i =1 X i is the sample average of independent random variables with ° = E ( X i ) and ± 2 = V ar ( X i ) for i = 1 ; 2 ; :::; n: Note that " ! d " denotes the "covergence in distribution", i.e. the distribution of p n ° ² X n ´ ° ³ becomes arbitrarily well approximated by the standard normal distribution N (0 ; 1) as n ! 1 : In order to apply the central limit theorem for the given random variable Z n ; dividing Z n by the standard deviation ± yields, Z n ± = 1 ± 1 p n ´ n P i =1 ( X i ´ ° ) µ = 1 ± 1 p n ´ n P i =1 X i ´ n ± ° µ = p n ± ´ 1 n n P i =1 X i ´ ° µ = p n ± ² X n ´ ° ³ ! d N (0 ; 1) by the central limit theorem where we °nd that Z n ° ! d Z with Z ~ N (0 ; 1) : Given this, we can calculate P ( Z n µ 0 : 924) as follows, P ( Z n µ 0 : 924) = P ° Z n ± µ 0 : 924 ± ± · P 0 @ Z n ± µ 0 : 924 q 2 9 1 A · P ( Z µ 1 : 96) · 0 : 975 where P ( Z µ 1 : 96) · 0 : 975 is found from the table 1 in the appendix (Given P ( Z µ ´ 1 : 96) = 0 : 025 from the table, we °nd P ( Z µ 1 : 96) = 1 ´ P ( Z ² 1 : 96) = 1 ´ 0 : 025 = 0 : 975 where P ( Z ² 1 : 96) = P ( Z µ ´ 1 : 96) = 0 : 025 due to the symmetry of the standard normal distribution.
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