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## Occupancy Problems

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**Occupancy Problems**m balls being randomly assigned to one of n bins. (Independently and uniformly) The questions: - what is the maximum number of balls in any bin? -what is the expected number of bins with k balls?**For arbitrary events: , not necessarily**independent: the probability of the union of events is no more than the sum of their probabilities. Let m=n : For let , where j is the number of balls in the th bin. Then we get: for all i.**Now: we concentrate on analyzing the 1st bin, so:**Let denote the event that bin has or more balls in it. So: From upper bound for binomial coefficients**Now, let**Then: with probability at least , no bin has more than balls in it!**The Birthday Problem**Now n=365, How large must m be before two people in the group are likely to share their birthday? For , let denote the event that the th ball lands in a bin not containing any of the first balls. But:**So:**Now we can see that for the probability that all m balls land in distinct bins is at most .**Markov Inequality**Let Y be a random variable assuming only non-negative values. Than for all : Or: Proof:define Than: Now, , , else**standard deviation**If X is a random variable with expectation , The variance is defined: The standard deviation of X is**Chebyshev’s Inequality**Let X be a random variable with expectation , and standard deviation . Then for any : Proof: First: for we get . applying the markov inequality to Y bounds this probability.