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Consider a user is uploading a set of files to a server. Assuming a bandwidth of $b$ bits/s, it takes $t_i=|f_i|/b$ seconds to upload the file, where $|f_i|$ is the size of file $i$. If the user does not report the file size to the server, from the server's perspective, the transmission time of the files, i.e, $t_i$, follow an exponential distribution. However, if the user report the file size before uploading the file, it doesn't follow exponential distribution anymore, because at least it doesn't have the memoryless property. How can such a situation be described?

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I think it depends on what the description is to be used for. As it is, I can't see any mathematical contents in this question. – Harald Hanche-Olsen Jan 25 '13 at 18:22
1. If t is the product of b and f, the units don't make sense. t is in seconds, b in bits/s and f in bits. 2. If the server does not know the size of the file, the time for additional s seconds is independent of how many seconds the upload has been going on for. It is memoryless. If on the other hand, the server knows the file size is say, 2 units and the transfer takes place at 1 units/sec, it will take 2 seconds plus some random noise for the upload to finish. – Inquest Jan 25 '13 at 18:34
@Inquest: thanks, 1. I fixed the unit's issue. 2. You description is correct and I mean the same. But I wonder how is it possible? The user transfers the same set of files in the two cases, but one case follows exponential distribution and the other one doesn't! – Helium Jan 25 '13 at 18:51
Exponential distribution is known to be a little counter intuitive when it comes to its memory less property. The best you can do to understand this is to look at more examples in a good Stochastic Models textbook. When I learnt it for the first time even I was flustered how such things could be possible but the more your practice and the more you see, you realize it actually makes sense. Its hard to explain but easy to realize on your own. – Inquest Jan 25 '13 at 19:04

The distribution of an unknown quantity often changes depending on what information is available. This is an example of marginal and conditional distributions: without any knowledge of the file sizes, the marginal distribution of the time required to transfer each file is exponential. Conditional on knowing the file sizes, though, the distribution of time required changes drastically.

Here's another example: Suppose I tell you that I chose a point x somewhere on the globe, and I ask you to give me a probability distribution for the temperature at that location at noon local time tomorrow. Without knowing where that point is, your distribution will be fairly wide, with values from below freezing to up to maybe 45 degrees celsius--the location might be in either Alaska or Australia, after all. But if I then tell you that the location is in Australia, the distribution conditional on that information is much different--much more mass near 40 degrees and none below freezing.

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