Say we have a sequence of n binary random variables A1, A2, ... An. How many...
9. In many cases where a sequence of random variables converges in probability to some b, this b will be either the expected value or the limit of the expected values of the variables. However, this is not generally true. (a) Consider a sequence of random variables where for each n, xn comes from this distribution with P(Xn = n) = 1/n and P(Xn = 0) = 1 - 1/n. Find limn+ E(Xn). (b) Find the value b such that...
(10 marks) Let X1, X2,... be a sequence of independent and identically distributed random variables with mean EX1 = i and VarX1 = a2. Let Yı, Y2, ... be another sequence of independent and identically distributed random variables with mean EY = u and VarY1 a2 Define the random variable ( ΣxΣ) 1 Dn 2ng2 i= i=1 Prove that Dn converges in distribution to a standard normal distribution, i.e., prove that 1 P(Dn ) dt 2T as n >oo for...
(1) Let a (.. ,a-2, a-1,ao, a1, a2,...) be a sequence of real numbers so that f(n) an. (We may equivalently write a = (abez) Consider the homogeneous linear recurrence p(A)/(n) = (A2-A-1)/(n) = 0. (a) Show ak-2-ak-ak-1 for all k z. (b) When we let ao 0 and a 1 we arrive at our usual Fibonacci numbers, f However, given the result from (a) we many consider f-k where k0. Using the Principle of Strong Mathematical Induction slow j-,-(-1...
Let n be a positive integer. We sample n numbers a1, a2,..., an from the set {1,...,n} uniformly at random, with replacement. We say that picks i and j with are a match if ai = aj, i < j. What is the expected total number of matches? Use indicators.
how to generating the binary code by arithmetic coding 1. Assume there are four letters from an information source with probabilities as A1 0.5 A2 0.3 АЗ 0.1 0.1 A4 Generate the tag and find the corresponding gaps and binary values for each stage and the sequence of the symbols which are coded are ala3a2a4al (25 Marks) 1. Assume there are four letters from an information source with probabilities as A1 0.5 A2 0.3 АЗ 0.1 0.1 A4 Generate the...
(1) Consider the probability space 2 [0, 1. We define the probability of an event A Ω to be its length, we define a sequence random variables as follows: When n is odd Xn (u) 0 otherwise while, when n is even otherwise (a) Compute the PMF and CDF of each Xn (b) Deduce that X converge in distribution (c) Show that for any n and any random variable X : Ω R. (d) Deduce that Xn does not converge...
Say we have data xi, . . ,z,, which are independent and identically distributed normal random variables with mean μ and variance 100. How often does this interval cover 11, 20 Say we have data xi, . . ,z,, which are independent and identically distributed normal random variables with mean μ and variance 100. How often does this interval cover 11, 20
Problem 3 Let A be a "random point" that coincides with the point a1,a2,as or as with equal probabil- ities. d4 Let a random variable X be the first coordinate of the point A, a random variable Y be the scond coordinate of the point A. (a) Find the joint distribution of random variables X and yY (b) Find the marginal distribution of X and Y. (c) Find the cumulative distribution functions of X and Y (d) Compute E(X] and...
Sequence a1,a2……an consists of nonnegative integers. We want to determine the largest sum of such subsequences that do not contain two consecutive elements of the original sequence (For example if a3 is a member of the subsequence, then a2 and a4 cannot be contained in the subsequence ). Give an O(n) running time algorithms for this problem.