概率论 (Summer 2013)/Problem Set 1
Problem 1
(Due to J. von Neumann.)
- Suppose you are given a coin for which the probability of HEADS, say [math]\displaystyle{ p }[/math], is unknown. How can you use this coin to generate unbiased (i.e., [math]\displaystyle{ \Pr[\mbox{HEADS}]=\Pr[\mbox{TAILS}]=1/2 }[/math]) coin-flips? Give a scheme for which the expected number of flips of the biased coin for extracting one unbiased coin-flip is no more than [math]\displaystyle{ 1/(p(1-p)) }[/math].
- Devise an extension of the scheme that extracts the largest possible number of independent, unbiased coin-flips from a given number of flips of the biased coin.
Problem 2
(Due to D.E. Knuth and A. C-C. Yao.)
- Suppose you are provided with a source of unbiased random bits. Explain how you will use this to generate uniform samples from the set [math]\displaystyle{ S=\{0,\dots,n-1\} }[/math]. Determine the expected number of random bits required by your sampling algorithm.
- What is the worst-case number of random bits required by your sampling algorithm? Consider the case when [math]\displaystyle{ n }[/math] is a power of [math]\displaystyle{ 2 }[/math], as well as the case when it is not.
- Solve (1) and (2) when, instead of unbiased random bits, you are required to use as the source of randomness uniform random samples from the set [math]\displaystyle{ \{0,\dots,p-1\} }[/math]; consider the case when [math]\displaystyle{ n }[/math] is a power of [math]\displaystyle{ p }[/math], as well as the case when it is not.
Problem 3
(Due to D.R. Karger and R. Motwani.)
- 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.
- 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 than some positive constant.
Problem 4
(Due to R.M. Karp)
Consider a bin containing [math]\displaystyle{ d }[/math] balls chosen at random (without replacement) from a collection of [math]\displaystyle{ n }[/math] distinct balls. Without being able to see or count the balls in the bin, please devise a strategy to simulate random sampling with replacement from the original set of [math]\displaystyle{ n }[/math] balls. Notice that your only access to the balls is that you can sample without replacement from the bin.
Problem 5
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?