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Convergence of Probability

  1. Oct 13, 2009 #1
    1. The problem statement, all variables and given/known data

    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)


    2. Relevant equations



    3. The attempt at a solution

    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)
    [tex]P(|\frac{Yn}{n} - p |\geq \epsilon ) \leq \frac{p^2 q^2}{n^2 \epsilon^2} [/tex]

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

    Is this correct?

    Thank you.
     
  2. jcsd
  3. Oct 14, 2009 #2
    If you're trying to show that it converges in probability, then yes.
     
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