MLE Homework: Distance Measurement with Odometer Error

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Homework Help Overview

The problem involves measuring the distance from home to office using a car's odometer, which only records whole miles. The original poster describes the scenario where the odometer readings are subject to measurement error, specifically that the readings can vary due to the nature of the odometer's design. The goal is to find the maximum likelihood estimate (MLE) of the exact distance based on recorded odometer data.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the implications of the odometer's inability to reset and how it affects the interpretation of the readings. There is consideration of the error distribution and its independence, with questions about how to formulate the likelihood based on the discrete nature of the odometer readings and the continuous error.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of the problem and the implications of the assumptions made about the error distribution. Some guidance has been offered regarding the relationship between the actual distance and the likelihood of the odometer readings, but no consensus has been reached.

Contextual Notes

Participants are working under the assumption that the measurement errors are uniformly distributed and independent, and they are grappling with how to incorporate this into their likelihood calculations. The lack of a reset button on the odometer is a key point of discussion, influencing the readings obtained during multiple trips.

Yagoda
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Homework Statement


John wants to measure the distance from his home to his office, so he drives to work several times and measures the distance on his car's odometer. Unfortunately, the odometer records distance only to the nearest mile. (Johns odometer changes abruptly from one digit to the next. When the odometer displays a digit d, it is not possible to infer how close it is to becoming d + 1.) John drives the route ten times and records the following data from the odometer: 3, 3, 3, 4, 3, 4, 3, 3, 4, 3. Find the m.l.e. of the exact distance.


Homework Equations





The Attempt at a Solution


The way I'm interpreting the problem, it seems the odometer doesn't have a button to reset it to zero at the beginning of the trip so he has to calculate the distance each time by subtracting the beginning readout from the end readout. This means he could have traveled up to 1 mile further than the odometer says he did.

We are allowed to make some assumptions so I am assuming that each error is independent and uniformly distributed on [0,1).

I'm stumped as to how to find the likelihood. I'm trying to come up with a distribution to find the likelihood, but don't know how to begin since it seems like we have this continuous variable, the error, that's begin added to the distance, but we are only looking at it discretely.

Thanks.
 
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With a reset button, it would always show 3 miles.

We are allowed to make some assumptions so I am assuming that each error is independent and uniformly distributed on [0,1).
A distance of 3.8 miles can give 3 as measurement.
 
Yeah, that's why I assumed there was no reset button, otherwise he would get the same readout each time.

A distance of 3.8 miles can give 3 as measurement.
Exactly, this what I'm having trouble encapsulating in the distribution of the odometer readout.
 
The distribution of the error will depend on the actual distance. This is useful, as it will give different likelihoods later.
 

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