hw6 - f Y ( v ) = 1 + v,-1 v , v, < v 1 , ,...

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EE 351K Probability, Statistics, and Random Processes FALL 2010 Instructor: Haris Vikalo hvikalo@ece.utexas.edu Homework 6 Due on : Tuesday 10/19/10 Problem 1: Ceiling of an Exponential : X is an exponential random variable with paramter λ . Y = ceil( X ) , where the ceiling function ceil( · ) rounds its argument up to the closest integer, i.e.: ceil( a ) = ± a if a is an integer the smallest integer bigger than a if a is not an integer What is the PMF of Y ? Is it one of the common random variables? (Hint: for all k , find the quantity P ( Y > k ) . Then find the PMF) Problem 2 : Gaussian Coordinates : A random point ( X,Y ) on a plane is chosen as follows: X and Y are chosen independently, with each one being a Gaussian random variable with zero mean and variance of 1. Let D be the square of the (random) distance of the point from the origin. Find the PDF of D . Is D one of the common random variables? Problem 3 : Let Y be a random variable with probability density function (PDF)
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Unformatted text preview: f Y ( v ) = 1 + v,-1 v , v, < v 1 , , otherwise . Find 1. P ( | Y | < 1 2 ) 2. P ( Y > | Y < 1 2 ) 3. E [ Y ] . Problem 4 : Let X,Y,Z be independent uniformly distributed random variables on [0 , 1] . Let W = max( X,Y,Z ) and R = min( X,Y,Z ) . Find the pdf of W , the pdf of R , and the joint pdf of ( W,R ) . Problem 5 : A signal s = 3 is transmitted from a satellite but is corrupted by noise, and the received signal is X = s + W . When the weather is good, which happens with probability 2 3 , W is normal with zero mean and variance 4 . When the weather is bad, W is normal with zero mean and variance 9 . In the absence of any weather information: 1. What is the PDF of X ? 2. Calculate the probability that X is between 2 and 4 ....
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