Let X₁, X₂.....X₁ be a random sample from an exponentialdistribution with parameter 8. Find the MLE for P(X≥1).

Answered: Let X₁, X₂,,... Xn be a random sample…

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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Question

Let X₁, X₂.....X₁ be a random sample from an exponential
distribution with parameter 8. Find the MLE for P(X≥1).

Expert Solution

Step 1: Define the Likelihood Function

Given a random sample $X subscript 1 comma X subscript 2 comma horizontal ellipsis comma X subscript n$ from an exponential distribution with parameter $θ$ , the probability density function (pdf) for each $X_{i}$ is:

$f open parentheses x subscript i ∣ theta close parentheses equals theta e to the power of negative theta x subscript i end exponent text for end text x subscript i greater or equal than 0 comma theta greater than 0$

The likelihood function $calligraphic L left parenthesis theta right parenthesis$ is the product of the individual pdfs:

$calligraphic L left parenthesis theta right parenthesis equals product from i equals 1 to n of f open parentheses x subscript i ∣ theta close parentheses equals product from i equals 1 to n of theta e to the power of negative theta x subscript i end exponent equals theta to the power of n e to the power of negative theta sum from i equals 1 to n of x subscript i end exponent$

To simplify calculations, it's common to work with the log-likelihood:

$log calligraphic L left parenthesis theta right parenthesis equals n log theta minus theta sum from i equals 1 to n of x subscript i$