随机算法 (Spring 2014)/Moments and Deviation

Problem 1

(Due to D.R. Karger and R. Motwani.)

1. Let [math]\displaystyle{ S,T }[/math] be two disjoint subsets of a universe [math]\displaystyle{ U }[/math] such that [math]\displaystyle{ |S|=|T|=n }[/math]. Suppose we select a random set [math]\displaystyle{ R\subseteq U }[/math] by independently sampling each element of [math]\displaystyle{ U }[/math] with probability [math]\displaystyle{ p }[/math]. We say that the random sample [math]\displaystyle{ R }[/math] is good if the following two conditions hold: [math]\displaystyle{ R\cap S=\emptyset }[/math] and [math]\displaystyle{ R\cap T\ne\emptyset }[/math]. Show that for [math]\displaystyle{ p=1/n }[/math], the probability that [math]\displaystyle{ R }[/math] is good is larger than some positive constant.
2. Suppose now that the random set [math]\displaystyle{ R }[/math] is chosen by sampling the elements of [math]\displaystyle{ U }[/math] with only pairwise independence. Show that for a suitable choice of the value of [math]\displaystyle{ p }[/math], the probability that [math]\displaystyle{ R }[/math] is good is larger than some positive constant.

Problem 2

Let [math]\displaystyle{ X_1,X_2,\dots,X_n }[/math] be independent geometrically distributed random variables each having expectation 2 (each of the [math]\displaystyle{ X_i }[/math] is an independent experiment counting the number of tosses of an unbiased coin up to and including the first HEADS). Let [math]\displaystyle{ X=\sum_{i=1}^nX_i }[/math] and [math]\displaystyle{ \delta }[/math] be a positive real constant. Derive the best upper bound you can on [math]\displaystyle{ \Pr[X\gt (1+\delta)(2n)] }[/math].