n17 - CS 70 Fall 2010 Discrete Mathematics and Probability...

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

View Full Document Right Arrow Icon
CS 70 Discrete Mathematics and Probability Theory Fall 2010 Tse/Wagner Lecture 17 Polling and the Law of Large Numbers Polling Question: We want to estimate the proportion p of Democrats in the US population, by taking a small random sample. How large does our sample have to be to guarantee that our estimate will be within (say) 0 . 1 of the true value with probability at least 0.95? This is perhaps the most basic statistical estimation problem, and it shows up everywhere. We will develop a simple solution that uses only Chebyshev’s inequality. More refined methods can be used to get sharper results. Let’s denote the size of our sample by n (to be determined), and the number of Democrats in it by the random variable S n . (The subscript n just reminds us that the r.v. depends on the size of the sample.) Then our estimate will be the value A n = 1 n S n . Now as has often been the case, we will find it helpful to write S n = X 1 + X 2 + ··· + X n , where X i = ( 1 if person i in sample is a Democrat; 0 otherwise. Note that each X i can be viewed as a coin toss, with Heads probability p (though of course we do not know the value of p ). And the coin tosses are independent. 1 Hence, S n is a binomial random variable with parameters n and p . What is the expectation of our estimate? E ( A n ) = E ( 1 n S n ) = 1 n E ( S n ) = 1 n × ( np ) = p . So for any value of n , our estimate will always have the correct expectation p . [Such a r.v. is often called an unbiased estimator of p .] Now presumably, as we increase our sample size n , our estimate should get more and more accurate. This will show up in the fact that the variance decreases with n : i.e., as n increases, the probability that we are far from the mean p will get smaller. To see this, we need to compute Var ( A n ) . But A n = 1 n S n , which is just a constant times a binomial random variable. Theorem 17.1 : For any random variable X and constant c, we have Var ( cX ) = c 2 Var ( X ) .
Image of page 1

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

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern