Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

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Solutions to Homework #6, 36-220 Due 12 October 2005 Non-Devore Problem The rule for variances which says Var ( X + Y ) = Var ( X )+Var ( Y ) only applies if X and Y are uncorrelated random variables, for instance, if X and Y are independent. But here the variables we’re adding are perfectly correlated — they’re functions of one another — and we can’t use that rule. (Notice that the book gives a more general rule, for correlated variables, as Equation 5.11 on page 244. You can check that this rule gives the right answer, once you calculate the covariance of X and Y .) 46 a E X = E [ X ] = 12cm σ X = σ X / n = 0 . 04cm / 16 = 0 . 01cm b E X = 12cm, again σ X = σ X / 64 = 0 . 04cm / 8 = 0 . 005cm c The larger sample is more likely to be close to 12 cm, because 12 cm is the mean, in both cases, but the standard deviation is smaller for the large sample. A smaller standard deviation implies a higher probability of being near the mean. 48 Notice that σ X = σ X / 100 = 1 / 10 = 0 . 1. a E X = 50, so Pr 49 . 75 - 50 0 . 1 X - 50 0 . 1 50 . 25 - 50 0 . 1 1
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= Pr (( - 0 . 25)(10) Z (0 . 25)(10)) = Φ(2 . 5) - Φ( - 2 . 5) = 0 . 988 b We repeat the same calculation, only substituting 49 . 8 for 50. Pr 49 . 75 - 49 . 8 0 . 1 X - 49 . 8 0 . 1 50 . 25 - 49 . 8 0 . 1 = Pr (( - 0 . 05)(10) Z (0 . 45)(10)) = Φ(4 . 5) - Φ( - 0 . 5) = 0 . 691 49 Let’s use T for the total time it takes to grade. T = 40 i =1 X i , where the X i are independent with E [ X i
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