Results 1 to 5 of 5

Math Help - Convergence of Probability

  1. #1
    Member
    Joined
    Oct 2009
    Posts
    218

    Convergence of Probability

    Let the random variable Yn have the distribution b(n,p).

    a)Prove that Yn/n converges in probability p.

    b)Prove that 1 - Yn/n converges to 1 - p.

    c)Prove that (Yn/n)(1 - Yn/n) converges in probability to p(1-p)

    So I need to use Chebyshev's inequality to solve it. E[Yn/n] = (1/n)*E[Yn] = (1/n)*(np) = p

    Var[Yn/n] = (1/n^2)*Var(Yn) =(1/n^2)*(npq) = pq/n

    a)
    P(|\frac{Yn}{n} - p |\geq \epsilon ) \leq \frac{p^2 q^2}{n^2 \epsilon^2}

    and as n approaches infinity \frac{p^2 q^2}{n^2 \epsilon^2} = 0 therefore Yn converges to p.

    Is this correct?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    Almost, the idea is correct, BUT you're off by a factor of n and the powers of p and q.
    The problem is you skipped a step and made 2 errors.

     P(|{Y_n\over n} - p |\geq \epsilon ) = P(|Y_n -np |\geq n\epsilon ) =P(|Y_n -np |^2\geq n^2\epsilon^2 )

    NOW use cheby's or markoff or whatever you want to call it

     \le {V(Y_n)\over n^2\epsilon^2} = {npq\over n^2\epsilon^2} = {pq\over n\epsilon^2}\to 0

    (b) follows from many results on convergence in probability
    OR use the same exact proof. The absolute value kills the minus sign, so there is no distinction...

     P\Biggl(\biggl| \biggl(1-{Y_n\over n}\biggr) -\biggl(1- p\biggr)\Biggr|\geq \epsilon \Biggr) =P(|{Y_n\over n} - p |\geq \epsilon ) = P(|Y_n -np |\geq n\epsilon ) ....

    (c) can be done several ways.

    You can just quote that {Y_n\over n} \to p hence \Biggl({Y_n\over n}\biggr)^2 \to p^2

    So the difference {Y_n\over n} -\Biggl({Y_n\over n}\biggr)^2 \to p-p^2

    Or you can use the rule about products.
    Last edited by matheagle; October 13th 2009 at 10:31 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Hello,

    For question a)... just another way round
    We can say that since Yn is a binomial distribution, it can be written as the sum of n independent rv's following a Bernoulli distribution(p), and then use the Law of Large Numbers, which will say that Yn/n converges to p almost surely.
    And it's well known that the almost sure convergence implies the convergence in probability :P
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Oct 2009
    Posts
    218
    Thank you both for your help/suggestion. I was wondering if either one of you knew a website that has a few more practice problems about this topic; my textbook only has 3 of them and I would like to get some practice at this.

    Thank you.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    the simplest book is
    Probability theory / R. G. Laha and V. K. Rohatgi
    I'm sure you can find it in most libraries
    Chung is ok too, but Laha and Rohatgi is the easiest
    I looked last night for more of this, for you
    But came up empty
    I then thought that you should get a copy of this book instead.
    It's basic and it has some nice examples.

    they even have it down under
    http://catalogue.nla.gov.au/Record/2535456
    No idea why it doesn't fall upward down there
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Convergence in Probability
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: October 8th 2011, 08:04 PM
  2. Convergence in probability but no convergence in L^p nor A.S.
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: April 16th 2011, 05:17 AM
  3. probability convergence
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: November 24th 2010, 11:49 PM
  4. fundamental in probability & convergence with probability 1
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: February 23rd 2010, 09:58 AM
  5. Almost sure convergence & convergence in probability
    Posted in the Advanced Statistics Forum
    Replies: 9
    Last Post: November 27th 2009, 11:31 PM

Search Tags


/mathhelpforum @mathhelpforum