{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

hw4sol

# hw4sol - STATS 116 Kihwan Choi Homework 4 solutions 1(3.3.9...

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

STATS 116 Kihwan Choi Homework 4 solutions 1. (3.3.9) Out of n individual voters at an election, r vote Republican and n - r vote Democrat. At the next election the probability of a Republican switching to vote Democrat is p 1 , and of a Democrat switching is p 2 . Suppose individuals behave independently. Find (a) the expectation and (b) the variance of the number of Republican votes at the second election. Solution: (a) Let X denote the number of Republican votes at the second election. Also X 1 and X 2 denote the number of Republican votes at the second election with Republican votes at the first election and the number of Republican votes at the second election with Democrat votes at the first election, respectively. Then X = X 1 + X 2 where X 1 B ( r, 1 - p 1 ) and X 2 B ( n - r, p 2 ). Thus, EX = EX 1 + EX 2 = r (1 - p 1 ) + ( n - r ) p 2 (b) Since X 1 and X 2 are independent, V ar ( X ) = V ar ( X 1 ) + V ar ( X 2 ) = rp 1 q 1 + ( n - r ) p 2 q 2 2. (3.3.12) A random variable X has expectation 10 and standard deviation 5. (a) Find the smallest upper bound you can for P ( X 20). Solution: By one-sided Chebyshev’s inequality (Cantelli’s inequality), P ( X 20) = P ( X - μ σ 2) 1 1 + 2 2 = 0 . 2 , which can be achievable when P ( X ) = 0 . 2 for X = 20 , 0 . 8 for X = 7.5 , resulting in μ = 10 and σ = 5. (b) Could X be a binomial random variable? Solution: No, it is impossible. If it is, we should have np = μ = 10 and npq = σ 2 = 25. This implies q = 2 . 5, but q = 1 - p should be in [0 1].

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