概率论 (Summer 2013)/Problem Set 2

Problem 1

We play the following game:

Start with [math]\displaystyle{ n }[/math] people, each with 2 hands. None of these hands hold each other. At each round, uniformly pick 2 free hands and let these two hands hold together. Repeat this until no free hands left.

• What is the expected number of cycles made by people holding hands with each other (one person with left hand holding right hand is also counted as a cycle) at the end of the game?

Problem 2

In Balls-and-Bins model, we throw [math]\displaystyle{ n }[/math] balls independently and uniformly at random into [math]\displaystyle{ n }[/math] bins, then the maximum load is [math]\displaystyle{ \Theta(\frac{\ln n}{\ln\ln n}) }[/math] with high probability.

The two-choice paradigm is another way to throw [math]\displaystyle{ n }[/math] balls into [math]\displaystyle{ n }[/math] bins: each ball is thrown into the least loaded of 2 bins chosen independently and uniformly at random and breaks the tie arbitrarily. The maximum load of two-choice paradigm is [math]\displaystyle{ \Theta(\ln\ln n) }[/math] with high probability, which is exponentially less than the previous one. This phenomenon is called the power of two choices.

Now consider the following three paradigms:

1. The first [math]\displaystyle{ n/2 }[/math] balls are thrown into bins independently and uniformly at random. The remaining [math]\displaystyle{ n/2 }[/math] balls are thrown into bins using two-choice paradigm.
2. The first [math]\displaystyle{ n/2 }[/math] balls are thrown into bins using two-choice paradigm. The remaining [math]\displaystyle{ n/2 }[/math] balls are thrown into bins independently and uniformly at random.
3. Assume all [math]\displaystyle{ n }[/math] balls are in a sequence. For every [math]\displaystyle{ 1\le i\le n }[/math], if [math]\displaystyle{ i }[/math] is odd, we throw [math]\displaystyle{ i }[/math]th ball into bins independently and uniformly at random, otherwise, we throw it into bins using two-choice paradigm.

What is the maximum load with high probability in each of three paradigms. You need to give an asymptotically tight bound (i.e. [math]\displaystyle{ \Theta(\cdot) }[/math]).

Problem 3

Consider a sequence of [math]\displaystyle{ n }[/math] flips of an unbiased coin. Let [math]\displaystyle{ H_i }[/math] denote the absolute value of the excess of the number of HEADS over the number of TAILS seen in the first [math]\displaystyle{ i }[/math] flips. Define [math]\displaystyle{ H=\max_i H_i }[/math]. Show that [math]\displaystyle{ \mathbf{E}[H_i]=\Theta(\sqrt{i}) }[/math], and that [math]\displaystyle{ \mathbf{E}[H]=\Theta(\sqrt{n}) }[/math].

Problem 4

Let [math]\displaystyle{ X }[/math] be a random variable with expectation [math]\displaystyle{ \mu_X }[/math] and standard deviation [math]\displaystyle{ \sigma_X{\color{red}\gt 0} }[/math].

1. Show that for any [math]\displaystyle{ t\in\mathbb{R}^+ }[/math],

   [math]\displaystyle{ \Pr[X-\mu_X\ge t\sigma_X]\le\frac{1}{1+t^2}, }[/math]

   This version of the Chebyshev inequality is sometimes referred to as the Chebyshev-Cantelli bound.

2. Prove that

   [math]\displaystyle{ \Pr[|X-\mu_X|\ge t\sigma_X]\le\frac{2}{1+t^2}. }[/math]

   Under what circumstances does this give a better bound than the Chebyshev inequality?

Problem 5 (Bonus Problem)

Consider the following experiment, which proceeds in a sequence of rounds. For the first round, we have [math]\displaystyle{ n }[/math] balls, which are thrown independently and uniformly at random into [math]\displaystyle{ n }[/math] bins. After round [math]\displaystyle{ i }[/math], for [math]\displaystyle{ i\ge 1 }[/math], we discard every ball that fell into a bin by itself in round [math]\displaystyle{ i }[/math] (i.e., we discard a ball if and only if there is no other balls that fell into the same bin). The remaining balls are retained for round [math]\displaystyle{ i+1 }[/math], in which they are thrown independently and uniformly at random into the [math]\displaystyle{ n }[/math] bins. Show that there is a constant [math]\displaystyle{ c }[/math] such that with probability [math]\displaystyle{ 1-o(1) }[/math], the number of rounds is at most [math]\displaystyle{ c\ln\ln n }[/math].