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?.
m balls being randomly assigned to one of n bins.
(Independently and uniformly)
- 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:
with probability at least , no bin has more than balls in it!
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.
Now we can see that for the probability that all m balls land in distinct bins is at most .
Let Y be a random variable assuming only non-negative values. Than for all :
standard deviation independent:
If X is a random variable with expectation ,
The variance is defined:
The standard deviation of X is
Let X be a random variable with expectation , and standard deviation . Then for any :
for we get .
applying the markov inequality to Y bounds this probability.