{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

asst7modified_shiehnr

# asst7modified_shiehnr - York University MATH 2030...

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

York University MATH 2030 3.0 (Elementary Probability) Assignment 7 – Solutions, Nov 2011 § 3.3 No. 3 (a) E [2 X + 3 Y ] = 2 E [ X ] + 3 E [ Y ] = 2 × 1 + 3 × 1 = 5 (b) Var[2 X + 3 Y ] = Var[2 X ] + Var[3 Y ] by independence. This then equals 2 2 Var[ X ] + 3 2 Var[ Y ] = 4 × 2 + 9 × 2 = 26. (c) By independence E [ XY Z ] = E [ X ] E [ Y ] E [ Z ] = 1 × 1 × 1 = 1. (d) Var[ XY Z ] = E [( XY Z ) 2 ] - E [ XY Z ] 2 = E [ X 2 Y 2 Z 2 ] - 1 2 by part (c). By independence this equals E [ X 2 ] E [ Y 2 ] E [ Z 2 ] - 1. But E [ X 2 ] = Var[ X ] + E [ X ] 2 = 2 + 1 2 = 3 and the same is true for the other two variables. Therefore Var[ XY Z ] = 3 × 3 × 3 - 1 = 26 . § 3.3 No. 14 (a) Let X be the income of a family picked uniformly at random from the given area. So E [ X ] = 10 , 000. We are asked to estimate the percentage of families with incomes over 50,000, or in other words to estimate P ( X > 50 , 000). Since X 0 we can use Markov’s inequality, and get P ( X > 50 , 000) E [ X ] 50 , 000 = 10 , 000 50 , 000 = 1 5 = 0 . 2 In other words, at most 20% of these families have income over \$50,000. (b) 50 , 000 = μ + 5 σ where μ = E [ X ] = 10 , 000 and σ = 8 , 000.

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern