Suppose that, for some N, the continuous random variables X₁, X2,..., XN are independently drawn from the uniform distribution on [−1,1]. That is, each Xį ~ Uniform([−1,1]). Let Sn = X₁ + X2 + ... + XN denote the sum of these uniform random variables. The Central Limit Theorem allows us to approximate the true sum Sy as a Gaussian random variable we'll call G. (It is up to you to find the mean and variance of G.) Notice that since each X₂ is bounded, Sy is also bounded. In particular, the true sum SN can only take values between -N and N. However, the Gaussian approximation to Sy is not bounded: G has some small probability of taking values greater than N (or less than -N). Find an expression for P(|G| > N). Your answer should be in terms of N, and it should involve the Gaussian cdf function (). Hint: for large values of N, P(|G| > N) goes to 0.

Suppose that, for some N, the continuous random variables X₁, X2,..., XN are independently drawn from the uniform distribution on [−1,1]. That is, each Xį ~ Uniform([−1,1]). Let Sn = X₁ + X2 + ... + XN denote the sum of these uniform random variables. The Central Limit Theorem allows us to approximate the true sum Sy as a Gaussian random variable we'll call G. (It is up to you to find the mean and variance of G.) Notice that since each X₂ is bounded, Sy is also bounded. In particular, the true sum SN can only take values between -N and N. However, the Gaussian approximation to Sy is not bounded: G has some small probability of taking values greater than N (or less than -N). Find an expression for P(|G| > N). Your answer should be in terms of N, and it should involve the Gaussian cdf function (). Hint: for large values of N, P(|G| > N) goes to 0.

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter8: Sequences, Series, And Probability

Section8.7: Probability

Problem 39E: Assume that the probability that an airplane engine will fail during a torture test is 12and that...

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