{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Lecture_14 - 1 LAW OF LARGE NUMBERS Lecture 14...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
1 LAW OF LARGE NUMBERS Lecture 14 ORIE3500/5500 Summer2009 Chen Class Today Law of Large Numbers Convergence Normal (Gaussian) Distribution 1 Law of Large Numbers The law of large numbers or l.l.n. is one of the most important theorems in probability and is the backbone of most statistical procedures. Theorem. If X 1 , . . . , X n are independent and identically distributed(iid) with mean μ , then the sample mean ¯ X n converges to the true mean μ as n in- creases, that is, ¯ X n -→ μ, n → ∞ . Before we try to see why we should expect this let us recall a few properties of the the sample mean, ¯ X n = X 1 + · · · + X n n . 1. Expected value of the sample mean, E ( ¯ X n ) = E X 1 + · · · + X n n · = 1 n E ( X 1 + · · · + X n ) = 1 n nE ( X 1 ) = μ. 1
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
2 NOTIONS OF CONVERGENCE 2. If var ( X 1 ) = σ 2 , then variance of the sample mean, var ( ¯ X n ) = var X 1 + · · · + X n n · = 1 n 2 var ( X 1 + · · · + X n ) = 1 n 2 ( var ( X 1 ) + · · · + var ( X n )) (by independence) = 1 n 2 n · var ( X 1 ) = σ 2 n . This means that the variance of the sample mean decreases as the sample size increases. Recall that the variance of a random variable measures the dispersion of the random variable about its mean. So if the variance is decreasing to 0, then the random variable is slowly shrinking to its mean. It becomes more and more concentrated around the population mean. The Chebyshev’s inequality completes the argument. For any ² > 0 P [ | ¯ X n - μ | > ² ] var ( ¯ X n ) ² 2 = σ 2 /n ² 2 0 , n → ∞ . This shows that whatever small number positive ² we choose, the probability that the sample mean is more than ² distance away from the true mean goes to zero. So we proved the LLN in the case when the variance of X 1 is finite. It can be proved without this assumption as well. Note that the statement of LLN does not assume anything about the variance of X 1 .
Image of page 2
Image of page 3
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