# Convergence of Probability

1. Oct 13, 2009

### cse63146

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

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

Is this correct?

Thank you.

2. Oct 14, 2009

### e(ho0n3

If you're trying to show that it converges in probability, then yes.