CMPSC 497股票市场模型

股票市场的一个简单模型如下：价格为q的股票将以因子r>1增加到qr概率p，并且将以概率1下降到q/r？p.如果我们从价格为1的股票开始，找到T天后股票价格的预期值和方差。问题2。紧密集中（a） 给定一个正整数k，描述一个随机变量X，假设只有非负值，使得Markov不平等现象十分严重。也就是说，使得Pr[X]≥a]=E[X]/a。（b） 给定一个正整数k，描述一个随机变量X，使得切比雪夫不等式是紧的。也就是说，这样Pr[|E？E[X]|≥k√Var（X）]=1/k2。问题3。又一个Chernoff绑定。设X1，Xn是n个独立随机变量，使得对于每i＝1，n.设Sn=∑ni=iXi，设μ=E[Sn]。表明对于任何δ>0Pr.CMPSC 497: Advanced Algorithms3 at 10:0

0 Problem

Set 2Notice: Type your answers using LaTeX and make sure to upload the answer file on Gradescope before the deadline.For more details and instructions, read the syllabus.Problem 1. StocksA simple model for the stock market works as follows: a stock with price q will increase by a factor r > 1 to qr withprobability p and will fall to q/r with probability 1? p. If we start with a stock with price 1, find formulae for theexpected value and variance of the price of the stock after T days.Problem 2. Tight concentration(a) Given a positive integer k, describe a random variable X assuming only non-negative values such that Markov’sinequality is tight. That is, such that Pr[X ≥ a] = E[X ]/a.(b) Given a positive integer k, describe a random variable X such that Chebyshev’s inequality is tight. That is,suchthat Pr[|E?E[X ]| ≥ k√Var(X)] = 1/k2.Problem 3. Another Chernoff bound.Let X1, . . . ,Xn be n independent random variables such that Xi ∈ [ai,bi] for every i= 1, . . . ,n. Let Sn = ∑ni=iXi and letμ = E[Sn]. Show that for any δ > 0Pr[|Sn?μ| ≥ δ ]≤ 2exp(? 2δ2∑ni=1(bi?ai)2).Hint: Follow the prove approach from class for 0/
1 random variables. When it is time to use the information about theXi’s, use of the fact that if Y is a random variable such that E[Y ] = 0 andY ∈ [a,b], then for any t ≥ 0, E[etY ]≤ et2(b?a)28 .Problem 4. Balance partitioningGiven an n×n matrix A whose entries are either
0 or 1, we want to find a column vector b ∈ {?1,1}n thatminimizes∥Ab∥∞. (Recall that if Ab = c with c = (c1, . . . ,cn),then ∥Ab∥∞ = maxi=1,...,n |ci|.) Consider the following algorithmfor choosing b: each entry of b is ?1 otherwise. Show that for this choice of b, ∥Ab∥∞ =O(√n lnn) with probability 1?O(n?1).1Problem 5. A random geometric networkSuppose n points are placed uniformly at random in the unit square. Each point is then connected to the k closestpoints. Let G be the resulting random graph. Show that there exists a constant c > 0 (independent of n and k), suchthat when k ≥ c logn, Pr[G is connected]→ 1 as n→ ∞.Hint: Partition the unit square into small squares of area lognn . Show that in every small square there is at least onepoint with high probability. Then, use a Chernoff bound to show that with high probability the number of points indisc of radius√a lognn is at most b logn for suitable constants a,b > 0. Show that these properties imply that G isconnected.Problem 6. Reduction IGiven two sets, P and Q, of n points in Rn space, your task is to find x ∈ P and y ∈ Q such that ∥x? y∥2 is minimum.(a) Show how to do this in O(n3) time.(b) Show how to do this in time O(n2 logn) if it suffices to find to find x ∈ P and y ∈Q whose distance 1.001 times theminimum possible distance. (If the min distance is 0, you must find x= y.)Problem 7. Reduction IIGiven n data points in Rd , where d = n5, we want to find a subset of 4 data points whose sum vector has the smallestlength. Formally, find the 4 points x1,x2,x3,x4 ∈ Rd such that ∥x1+ x2+ x3+ x4∥
2 is minimized.(a) Show how to do this in O(n9) time.(b) Show how to do this in time O(n5 logn) if it suffices to find a subset whose sum has a length that is 1.001 the lengthof the smallest possible sum.
WX：codehelp