1. ## 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?

2. 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.

3. 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

4. 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.

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