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This exercise gives a less standard example of MLE. Let m be a positive integer, and U₁, U2,,Um are i.i.d. Uniform([0, 1]) RV's. Let X = max(U₁, U2, ···,Um), i.e. X is the highest outcome among m random numbers independently and uniformly chosen between 0 and 1. (i) What is the p.d.f. of the random variable X? (ii) Suppose we don't know the value of m and wish to determine it from an i.i.d. sample ,xn) of the random variable X. Verify that the log-likelihood function x = (x1, x2, equals l(x; m) = = n log m - n (m-1) log(1/x₂). i=1 (iii) Given x1,x2,...,xn, find a formula for the real number m* such that the function l above is maximized at m = m*. (However, m* is not the MLE estimator for m because it is not necessarily an integer.)

This exercise gives a less standard example of MLE. Let m be a positive integer, and U₁, U2,,Um are i.i.d. Uniform([0, 1]) RV's. Let X = max(U₁, U2, ···,Um), i.e. X is the highest outcome among m random numbers independently and uniformly chosen between 0 and 1. (i) What is the p.d.f. of the random variable X? (ii) Suppose we don't know the value of m and wish to determine it from an i.i.d. sample ,xn) of the random variable X. Verify that the log-likelihood function x = (x1, x2, equals l(x; m) = = n log m - n (m-1) log(1/x₂). i=1 (iii) Given x1,x2,...,xn, find a formula for the real number m* such that the function l above is maximized at m = m*. (However, m* is not the MLE estimator for m because it is not necessarily an integer.)

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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Can you show me the answer to (i)(ii) and (III). Thank you so much for your help

This exercise gives a less standard example of MLE. Let m be a positive integer, and U₁, U₂,,Um are i.i.d. Uniform([0, 1]) RV's. Let X = max(U₁, U₂,...,Um), i.e. X is the highest outcome among m random numbers independently and uniformly chosen between 0 and 1. (i) What is the p.d.f. of the random variable X? (ii) Suppose we don't know the value of m and wish to determine it from an i.i.d. sample x = (x₁, x2,,xn) of the random variable X. Verify that the log-likelihood function equals n l(x; m) = n log m - (m − 1) Σlog(1/x;). = i=1 (iii) Given £₁, X2, ‚¤n, find a formula for the real number m* such that the function l above is maximized at m m*. (However, m* is not the MLE estimator for m because it is not necessarily an integer.) (iv) Suppose we are given the sample data : 0.95 0.73 0.87 0.72 0.95 0.68 0.88 0.85 0.99 0.79 0.8 0.94 0.94 0.91 0.91 0.87 Plot l(x; m) as a function of m for the given sample data ☎ and compute the value of m* (as defined in the previous part). (v) What is the positive integer m that maximizes the function for this sample? This is the MLE estimate of m for this sample¹. (Hint: you could eyeball the plot to get the answer. A more mathematical approach is to test which of the two nearest integers to m* would give a higher value of l.)
Transcribed Image Text:This exercise gives a less standard example of MLE. Let m be a positive integer, and U₁, U₂,,Um are i.i.d. Uniform([0, 1]) RV's. Let X = max(U₁, U₂,...,Um), i.e. X is the highest outcome among m random numbers independently and uniformly chosen between 0 and 1. (i) What is the p.d.f. of the random variable X? (ii) Suppose we don't know the value of m and wish to determine it from an i.i.d. sample x = (x₁, x2,,xn) of the random variable X. Verify that the log-likelihood function equals n l(x; m) = n log m - (m − 1) Σlog(1/x;). = i=1 (iii) Given £₁, X2, ‚¤n, find a formula for the real number m* such that the function l above is maximized at m m*. (However, m* is not the MLE estimator for m because it is not necessarily an integer.) (iv) Suppose we are given the sample data : 0.95 0.73 0.87 0.72 0.95 0.68 0.88 0.85 0.99 0.79 0.8 0.94 0.94 0.91 0.91 0.87 Plot l(x; m) as a function of m for the given sample data ☎ and compute the value of m* (as defined in the previous part). (v) What is the positive integer m that maximizes the function for this sample? This is the MLE estimate of m for this sample¹. (Hint: you could eyeball the plot to get the answer. A more mathematical approach is to test which of the two nearest integers to m* would give a higher value of l.)
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