MLE Homework: Distance Measurement with Odometer Error

[Yagoda]

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.

 

[mfb]

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.

 

[Yagoda]

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.

 

[mfb]

The distribution of the error will depend on the actual distance. This is useful, as it will give different likelihoods later.