The central limit theorem can be used to analyze round-off error. Suppose that the round-off error is represented as a uniform random variable on [-1 2, 1 2 ]. If 100 numbers are added, approximate the probability that the round-off error exceeds (a) 1, (b) 2, and (c) 5.

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Mathematics

Amanda Dunlap

17 January, 15:34

The central limit theorem can be used to analyze round-off error. Suppose that the round-off error is represented as a uniform random variable on [-1 2, 1 2 ]. If 100 numbers are added, approximate the probability that the round-off error exceeds (a) 1, (b) 2, and (c) 5.

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Gabriella Crane · Answer 1

The mean of the sum of 100 variables is 100*0 since all have mean 0. The variance is the sum of the variances, which is 100 * 1/12 = 8.333. The standard deviation is the square root, 2.887.

a. The probability that the sum is greater than 1 is Prob[x > 1] = Prob[ (x - 0) / 2.887 > (1 - 0) / 2.887] =.3645 If you interpret this to mean that the absolute value of the sum is > 1, then the probability is doubled.

b. Prob[x > 2] = Prob (z > 2/2.887) =.245 (or. 490).

c. Prob[x > 5] = Prob (z > 5/2.887) =.0416 (or. 0892).