1. This exercise addresses the question: How many observations do you need to coveran unknown percentile of a distribution?Suppose X; * F(-) for i = 1,2,...,n denote a random sample of size n from acontinuous cumulative distribution function (CDF); i.e., F(x) = Pr[X; < a] forany i = 1,2,...,n. Suppose we want to estimate the p-th quantile Q, = F-(p)%3Dfor some p e (0, 1). Determine the minimal sample size n required such that theobserved sample would cover the p-th quantile with at least 99% probability. Inother words, find smallest integer n such thatPr[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 inequalityabove dependsonly 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 and0.50 satisfying the inequality

In this problem, we are given a random sample of size n from a continuous cumulative distribution…

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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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Show all work. Do not reuse a former solution.
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6. Let X1, X2, ...X100 denote a random sample of size 100 from an exponential distribution with mean 1. Let Y equal the sum of all the X's. Approximate the probability that Y is greater than 110. |
Q1: Suppose the number of customers X that enter a store between the hours 9:00 a.m. and 10:00 a.m. follows a Poisson distribution with parameter 0. Suppose a random sample of the number of customers that enter the store between 9:00 a.m. and 10:00 a.m. for 10 days results in the values 9, 7, 9, 15, 10, 13, 11, 7, 2, 12 Determine the maximum likelihood estimate of 0. Show that it is an unbiased estimator. Q2: Assume that X is a discrete random variable with pmf f(x). Let X₁,...,X₁ be a random sample on X. Suppose that the space of X is finite, say, D={a₁,...,m}. An intuitive estimate of p(a) is the relative frequency of a, in the sample. We express this more formally as follows. For j=1, 2,..., m, define the statistics 1,(X) = {1 0 X₁ = a; X₁ = a; Then the intuitive estimate of p(a)) can be expressed by the sample average p(a) = -1,(X₂) Find the unbiased estimator and the variance of the estimator and its mgf.
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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;