MIT6_041F10_assn09

MIT6_041F10_assn09 - Massachusetts Institute of Technology...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Fall 2010) Problem Set 9 Due November 22, 2010 1. Random variable X is uniformly distributed between 1 . 0 and 1 . 0. Let X 1 ,X 2 ,... , be indepen- dent identically distributed random variables with the same distribution as X . Determine which, if any, of the following sequences (all with i = 1 , 2 ,... ) are convergent in probability. Fully justify your answers. Include the limits if they exist. X 1 + X 2 + ... + X i (a) U i = i (b) W i = max( X 1 ,...,X i ) (c) V i = X 1 X 2 ... X i 2. Demonstrate that the Chebyshev inequality is tight, that is, for every , > 0, and c , construct a random variable X with mean and standard deviation such that 2 P ( | X | c ) = 2 c Hint: You should be able to do this with a discrete random variable that takes on only 3 distinct values with nonzero probability. 3. Assume that a fair coin is tossed repeatedly, with the tosses being independent. We want to determine the expected number of tosses necessary to first observe a head directly followed by a tail. To do so, we define a Markov chain with states S,H,T,HT , where S is a starting state, H indicates a head on the current toss, T indicates a tail on the current toss (without heads on the previous toss), and HT indicates heads followed by tails over the last two tosses. This Markov indicates heads followed by tails over the last two tosses....
View Full Document

This note was uploaded on 01/11/2012 for the course EE 6.431 taught by Professor Prof.dimitribertsekas during the Fall '10 term at MIT.

Page1 / 4

MIT6_041F10_assn09 - Massachusetts Institute of Technology...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online