A Statistical Problem on Laptop Uptimes

Suppose you are in a large university campus. Most students here use laptops, and if you look around, you’d see most of them either working, listening to music or doing something else on their laptops. Suppose now you think of a quick project, of listing the uptimes of the laptops (how long they’ve been running). In Windows this is quite simple. Under the ‘Performance’ tab in Task Manager, you’ll notice that ‘Uptime’ gives the duration for which the laptop has been running. This timer, however, keeps counting from where it left off if you resume from hibernation, as it should, but we shall assume that no such cases happen in our campus.

Now, the campus is huge, and many students are using their laptops. You figure out some methodical way of visiting each student so that there are no over- or under-counting errors. But it takes you quite a while to visit all of them and note down their uptimes. So that if, for example, it took you six hours to collect all the data, then you took the last uptime reading six hours after the first.

If we assume a very large campus where the uptimes of the laptops of different students are completely independent of each other, the questions are the following:

1. On an average, is it going to make any difference to the statistics you collected, if instead of taking a long time to go around the campus, you could somehow acquire all of the uptimes at one instant of time?

2. Is the data you collected going to be distributed differently from that of the maximum uptimes of student laptops (duration before they shut it down)?

Unless I find myself without the time and effort, I plan to return and solve it in this blog post. (I haven’t solved it yet.)