Isye 2027

# The cases 1 u 0 and 0 u 1 will be considered

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Unformatted text preview: d variance, for sums of uniformly distributed random variables on [0, 1]. For n = 3 and n = 4, the Gaussian pdf is the one with a slightly higher peak. Solution: As noted in Example 2.2.7, the number showing for one roll of a die has mean µ = 3.5 and variance σ 2 = 2.9167. Therefore, S has mean 3500, variance 2916.7, and standard deviation √ 2916.7 = 54.01. By the Gaussian approximation with the continuity correction, P {|S − 3500| ≤ L} = P {|S − 3500| ≤ L + 0.5} S − 3500 L + 0.5 =P ≤ 54.01 54.01 L + 0.5 ≈ 1 − 2Q . 54.01 Now Q(1.645) = 0.05, so the desired value of L solves L + 0 .5 ≈ 1.645, 54.01 or L ≈ 88.84. Thus, L = 89 should give the best approximation. Example 4.9.6 Suppose each of 100 real numbers are rounded to the nearest integer and then added. Assume the individual roundoﬀ errors are independent and uniformly distributed over the interval [−0.5, 0.5]. Using the Gaussian approximation suggested by the CLT, ﬁnd the approximate probability that the absolute value of the sum of th...
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