1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Taking a Limit of a Probability

  1. Feb 10, 2014 #1
    1. The problem statement, all variables and given/known data I want to show that if I have a consistent sequence of estimators [itex]W_n[/itex] for [itex]\theta[/itex], i.e. [itex]\lim_{n \rightarrow \infty} P(|W_n - \theta| < \epsilon) = 1[/itex], then [itex]U_n = a_nW_n + b_n[/itex] is also a consistent sequence of estimators for [itex]\theta[/itex] where [itex]\lim_{n \rightarrow \infty}a_n = 1 [/itex] and [itex]\lim_{n \rightarrow \infty}b_n = 0. [/itex]

    2. Relevant equations

    3. The attempt at a solution
    We are looking at [itex]\lim_{n \rightarrow \infty} P(|a_nW_n + b_n - \theta| < \epsilon)[/itex], which is equivalent to [itex]\lim_{n \rightarrow \infty} P(-\epsilon < a_nW_n + b_n - \theta < \epsilon)[/itex] or [itex]\lim_{n \rightarrow \infty} P(-\epsilon - b_n< a_nW_n - \theta < \epsilon - b_n) [/itex].
    My question is how I can apply what I know about the limits of [itex]W_n, a_n, b_n[/itex] in this expression. While I would like to be able to say that since the an's go to 1 and the bn's to 0 I can apply the consistency of Wn, but I don't know if that is acceptable.
  2. jcsd
  3. Feb 10, 2014 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Use the definition of limit for the a and b sequences. Given ϵ > 0 there exists N such that...
  4. Feb 10, 2014 #3

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    [tex] \{-\epsilon - b_n < a_n W_n - \theta < \epsilon -b_n \}
    = \left\{ \frac{-\epsilon - b_n + \theta (1 -a_n)}{a_n}
    < W_n - \theta < \frac{ \epsilon - b_n + \theta (1-a_n)}{a_n} \right\}[/tex]
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted