Hmwk6sol2027 - Solutions to Homework 6 ISyE 2027 Fall 2006 Problem 1(a geometric(b Bernoulli(c binomial(d Poisson Problem 2(a We know that 1 = ∑

Info iconThis preview shows pages 1–3. 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: Solutions to Homework 6, ISyE 2027 Fall 2006 Problem 1 (a) geometric (b) Bernoulli (c) binomial (d) Poisson Problem 2 (a) We know that 1 = ∑ 4 k =1 P ( X = k ), so 1 = 4 X k =1 P ( X = k ) = c 4 X k =1 k = 10 c so c = 1 / 10. (b) The expected value (or mean) of X is just E [ X ] = 4 X k =1 kP ( X = k ) = (1 / 10) 4 X k =1 k 2 = 30 / 10 = 3 . (c) To compute the probability mass function of Y , we first need to determine the range of Y , or the set of possible values Y can take. It’s clear that the range of Y is { 1 , 4 , 9 , 16 } . Therefore, the probability mass function of Y is P ( Y = 1) = P ( X = 1) = 1 / 10 P ( Y = 4) = P ( X = 2) = 2 / 10 P ( Y = 9) = P ( X = 3) = 3 / 10 P ( Y = 16) = P ( X = 4) = 4 / 10 (d) The expected value of Y = X 2 is just E [ Y ] = (1) P ( Y = 1) + (4) P ( Y = 4) + (9) P ( Y = 9) + (16) P ( Y = 16) = 1 / 10 + 8 / 10 + 27 / 10 + 64 / 10 = 100 / 10 = 10 . (e) Using the “law of the unconscious statistician”, we see that E [ X 2 ] = 4 X k =1 k 2 P ( X = k ) = 4 X k =1 (1 / 10) k 3 = 10 . (f) Here E [ X ( X- 1)] = E [ X 2- X ] = E [ X 2 ]- E [ X ] = 10- 3 = 7 . (g) The variance of X is just V ar ( X ) = E [ X 2 ]- ( E [ X ]) 2 = 10- 9 = 1 . (h) The standard deviation is just the square root of the variance, so in this case it’s 1. (i) If we condition on the event that { X ≤ 2 } , then X is either 1 or 2. 1 2 P ( X = 1 | X ≤ 2) = P ( X = 1 ,X ≤ 2) P ( X ≤ 2) = P ( X = 1) P ( X ≤ 2) = (1 / 10) (3 / 10) = 1 / 3 ....
View Full Document

This note was uploaded on 02/01/2010 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.

Page1 / 5

Hmwk6sol2027 - Solutions to Homework 6 ISyE 2027 Fall 2006 Problem 1(a geometric(b Bernoulli(c binomial(d Poisson Problem 2(a We know that 1 = ∑

This preview shows document pages 1 - 3. Sign up to view the full document.

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