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1. This exercise addresses the question: How many observations do you need to cover an unknown percentile of a distribution? Suppose X, F(.) for i=1,2,...,n denote a random sample of size n from a continuous cumulative distribution function (CDF); i.e., F(x) = Pr[X;

1. This exercise addresses the question: How many observations do you need to cover an unknown percentile of a distribution? Suppose X, F(.) for i=1,2,...,n denote a random sample of size n from a continuous cumulative distribution function (CDF); i.e., F(x) = Pr[X;

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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1. This exercise addresses the question: How many observations do you need to cover an unknown percentile of a distribution? Suppose X; * F(-) for i = 1,2,...,n denote a random sample of size n from a continuous cumulative distribution function (CDF); i.e., F(x) = Pr[X; < a] for any i = 1,2,...,n. Suppose we want to estimate the p-th quantile Q, = F-(p) %3D for some p e (0, 1). Determine the minimal sample size n required such that the observed sample would cover the p-th quantile with at least 99% probability. In other words, find smallest integer n such that Pr[X4) < Q, < X(m)] 2 0.99, where X(1) = min{X1,..., X,} and X(m) = max{X1,... X,}. ..... (a) Show that the minimal sample size n that satisfies inequality above depends only on p (and not on the unknown CDF F). (b) Obtain the minimum value of n for p = 0.01,0.025, 0.05, 0.10, 0.20, 0.40 and 0.50 satisfying the inequality
Transcribed Image Text:1. This exercise addresses the question: How many observations do you need to cover an unknown percentile of a distribution? Suppose X; * F(-) for i = 1,2,...,n denote a random sample of size n from a continuous cumulative distribution function (CDF); i.e., F(x) = Pr[X; < a] for any i = 1,2,...,n. Suppose we want to estimate the p-th quantile Q, = F-(p) %3D for some p e (0, 1). Determine the minimal sample size n required such that the observed sample would cover the p-th quantile with at least 99% probability. In other words, find smallest integer n such that Pr[X4) < Q, < X(m)] 2 0.99, where X(1) = min{X1,..., X,} and X(m) = max{X1,... X,}. ..... (a) Show that the minimal sample size n that satisfies inequality above depends only on p (and not on the unknown CDF F). (b) Obtain the minimum value of n for p = 0.01,0.025, 0.05, 0.10, 0.20, 0.40 and 0.50 satisfying the inequality
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