Download On several Questions in Applied Probability

On several Questions in Applied Probability
We will give here several examples of questions we are interested in.
• For a positive integer n and 1 ≤ k ≤ n, what can be said about a k-wise independent
distribution on {0, 1}n ? More specifically, suppose the coordinates are given by
random variables Xi , 1 ≤ i ≤ n, where P (Xi = 1) = p and P (Xi = 0) = 1 − p for
an arbitrary fixed p ∈ (0, 1). What is the maximum (over all such distributions) of
P (X1 = X2 = ... = Xn = 1)? the minimum? same for other functions, such as the
entropy?
• Consider a combinatorial optimization problem, such as MAX-k-SAT or Vertex
Cover. Develop algorithms with good performance based not only on expectation
calculations, but also on the second moment (and possibly additional moments).
• Consider the following question, arising from Machine Learning Theory. We are
given a finite probability sample space with n atoms (of non-necessarily equal probabilities). We take a large independent sample from the space. Estimate the probability of the set of atoms that have been not picked even once in the sample. More
accurately, it is easy to calculate the expected unseen mass (as a function of the
atoms probabilities). Find good bounds on the probability of large deviations from
the expected value.
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