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Unformatted text preview: (c) By Bayes’ theorem, we find P ( I  D ) = P ( D  I ) P ( I ) P ( D ) = (1 . 04)(0 . 50) 1 . 034 = 0 . 4969 . 3. Let the probability mass function of X be given by f ( x ) = 2 x 1 16 , x = 1 , 2 , 3 , 4 . (10) (a) Find the cumulative distribution function F ( x ) of X . (10) (b) Find the mean, variance, and standard deviation of X . Solution . (a) It is easy to find that F (1) = f (1) = 1 / 16, F (2) = F (1) + f (2) = 1 / 16 + 3 / 16 = 1 / 4, F (3) = F (2) + f (3) = 1 / 4 + 5 / 16 = 9 / 16, and F (4) = F (3) + f (4) = 9 / 16 + 7 / 16 = 1. So, the cumulative distribution function is given by F ( x ) = , x < 1 1 16 , 1 ≤ x < 2 1 4 , 2 ≤ x < 3 9 16 , 3 ≤ x < 4 1 , x ≥ 4 . (b) μ = E ( X ) = 4 X x =1 xf ( x ) = 1 · (1 / 16) + 2 · (3 / 16) + 3 · (5 / 16) + 4 · (7 / 16) = 3 . 125 , E ( X 2 ) = 1 2 · (1 / 16) + 2 2 · (3 / 16) + 3 2 · (5 / 16) + 4 2 · (7 / 16) = 10 . 625 , σ 2 = E ( X 2 ) μ 2 = 10 . 625 3 . 125 2 = 0 . 859375 . σ = √ σ 2 = √ . 859375 = 0 . 927 . 4. The probability that your call to a service line is answered in less than 30 seconds is 0.60. Assume that your calls are independent. (5) (a) If you call 10 times, what is the probability that at least 6 calls are answered in less than 30 seconds? (5) (b) If you call 10 times, what is the mean number of calls that are answered in less than 30 seconds?...
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 Summer '08
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 Statistics, Probability, Probability theory, Florida Atlantic University, Hongwei Long

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