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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?.

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Occupancy problems
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 1 independent:st bin, so:

Let denote the event that bin has or more balls in it. So:

From upper bound for binomial coefficients


Now, let independent:

Then:

with probability at least , no bin has more than balls in it!


The birthday problem
The Birthday independent: 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: independent:

Now we can see that for the probability that all m balls land in distinct bins is at most .


Markov inequality
Markov Inequality independent:

Let Y be a random variable assuming only non-negative values. Than for all :

Or:

Proof:define

Than:

Now,

,

,

else


standard deviation independent:

If X is a random variable with expectation ,

The variance is defined:

The standard deviation of X is


Chebyshev s inequality
Chebyshev’s independent: 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.


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