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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 X₁ ~ 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 X; 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 Þ(.). 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 X₁ ~ 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 X; 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 Þ(.). Hint: for large values of N, P(|G| > N) goes to 0.

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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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 X₂ 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 SN 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 X, 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 Þ(.). Hint: for large values of N, P(|G| > N) goes to 0.
Transcribed Image Text: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 X₂ 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 SN 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 X, 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 Þ(.). Hint: for large values of N, P(|G| > N) goes to 0.
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