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cse63146
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Homework Statement
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)
Homework Equations
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.