Calculate the Expectation of Discrete Uniform Distribution

We have 1000 users. Now we have a new feature, which is applied to 10 new users each day starting day 1, until all the users see this feature on day 100. How many days does it take for an average user to wait and see this feature?

For any user, the probability of him being picked on any day n where 1\leq n \leq 100 is equal to the number of users picked on that date divided by total number of users, which is \frac{10}{1000} = \frac{1}{100}. This is a discrete uniform distribution.

The probability mass function of this distribution is f(n) = \frac{1}{100}, where n is the day number, and 1\leq n \leq 100.

The number of days it takes for an average user to wait and see this feature, is simply the expected day of this distribution, which can be calculated using E[\frac{n}{100}] = \sum_{n=1}^{100} n\cdot \frac{n}{100} = 1 \times \frac{1}{100} + 2 \times \frac{2}{100} + ... + 100 \times \frac{100}{100} = 50.5.


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