Three Problems On Inference

2017-12-24

Here I am proposing three likely related problems on Statistical Inference. As far as I can see, some of them can be solved by tools in Survival Analysis, some can be solved by doing a Bayesian Inference.

The Ruler Problem

This is classical. Suppose you have a ruler of length \(x\). However the pool of items you want to measure contains those with length more than \(x\). How would you make an inference to some summary statistics (the mean, for example) describing the lengths of the rulers in the pool?

The Train Problem

Suppose a very punctual train arrives per \(x\) minutes. You leave home every workday morning and record your waiting time until the train arrives. How would you estimate \(x\) from your data?

The Girlfriend Problem

Suppose the girlfriend meets the boyfriend every Friday evening at a place to go to dinner together. The two have a different distribution of arrival times, say \(F_b\) and \(F_g\). Sometimes the boyfriend arrives earlier, and sometimes the girlfriend arrives earlier. Suppose you are the boyfriend, and you only record the amount of time you wait, and the time you arrive, if you arrive earlier. And only the time you arrive if you arrive late. Now there may be a couple of sub-problems:

Suppose the boyfriend wants to propose and therefore definitely wants to arrive earlier than the girlfriend. How can he decide what time he might arrive to be very sure?
Suppose the boyfriend wants to make a good impression before he proposes and wants to arrive earlier than the girlfriend \(p\) percent of time. How would he decide what time to arrive base on his past notes?