This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Boston University Department of Electrical and Computer Engineering EC505 STOCHASTIC PROCESSES Problem Set No. 1 Fall 2008 Issued: Wednesday, Sept. 3, 2008 Due: Friday, Sept. 12, 2008 Problem 1.1 A random experiment consists of tossing a die and observing the number of dots on the top face. Let A 1 be the event that 3 comes up, A 2 the event that an even number comes up, and A 3 the event that an odd number comes up. (a) Find P ( A 1 ) , P ( A 1 A 3 ). (b) Find P ( A 2 A 3 ), P ( A 2 A 3 ), P ( A 1  A 3 ). (c) Are A 2 and A 3 disjoint? (d) Are A 2 and A 3 independent? Problem 1.2 A group of students is taking a multiplechoice test. For a particular question on the test, the fraction of students who know the answer is p . The fraction that will have to guess the answer is (1 p ). If a student knows the answer, then he will certainly answer the question correctly. If a student doesnt know the answer and must guess, then the probability of answering the question correctly is 1 /n , where n is the number of choices for the given question. (a) Compute the probability P c that a student who answers the question correctly actually knew the answer. (b) Suppose that the professor believes that p = . 85, i.e. that 85% of the students actually know the answer. Further, suppose that he wants to design the multiple choice question such that P c = . 95, i.e. so that correct answers on the question indicate actual knowledge with 95% probability. How many parts n should the problem have? Problem 1.3 A random variable x has probability distribution function P X ( x ) = 1 e 2 x u ( x ) where u ( ) is the unitstep function. (a) Calculate the following probabilities: P [ X 1] , P [ X 2] , P [ X = 2] . (b) Find p X ( x ), the probability density function for X . (c) Let Y be a random variable obtained from X as follows: Y = , if X < 2 1 , if X 2 Find p Y ( y ), the probability density function for Y . Problem 1.4 Let X and Y be statistically independent random variables with probability density functions p X ( x ) = 1 2 ( x + 1) + 1 2 ( x 1) , p Y ( y ) = 1 2 2 exp y 2 2 2 and let Z = X + Y , and W = XY . 1 (a) Find the conditional probability density functions p Z  X ( z  x = 1) and p Z  X ( z  x = 1)....
View
Full
Document
This note was uploaded on 09/29/2009 for the course EC 505 taught by Professor Karl during the Fall '04 term at BU.
 Fall '04
 Karl

Click to edit the document details