Statistics point estimators?

i just need some reasurance on the answers to some of these questions. I feel like i'm not getting the idea of point estimators, so if anyone could give me some tips/pointers or just trying to explain it. Thanks, Ok, here goes.





Let {X1,...Xn} be a random sample from Bernoulli distribution with P(X1=1)=PI=1-p(x1=0). Let PI^ (PI hat) = x-bar = (X1+...Xn)/n be an estimator for PI.


a) Find MSE(PI^). Is PI^ biased? Is PI^ consistent?


b) let Y= X1+...+Xn. Find the distribution of Y


c) Find the distribution of PI^ (which is called the sampling distribution of Pi^)





Okay, what i have for a). MSE = 0, Bias = 0 i.e, unbiased. and it is consistent.


b) i think that the distribution of Y is Binomial (sum of n-Bernoullis)


c) i think that when n is large enough, the distribution will be Normal.





However, i don't quite get where or not i can use the information from the population as if i were dealing with the sample on it's own i'd just have to make some guesses. Yeah, i'm quite confused.|||a) The MSE is only 0 as n -%26gt; infinity. To get the MSE you need to take the variance of the estimator, which will just be the sum of the individual variances divided by n^2 (when you take the variance of a random variable multiplied by a constant, you can move the constant outside the variance operator by squaring it). It works out to MSE{Phat} = Var{X}/n. This makes sense: the larger our sample, the smaller the MSE.





To determine if the estimator is biased, take the expected value. This is straightforward, and you get that Phat is, indeed, unbiased (remember that an estimator is unbiased if its expected value is the paramater that you are looking for).





Phat is consistent by the theorem that states that if an estimator theta is a) unbiased and b) Var{theta} -%26gt; 0 as n-%26gt;infinity then theta is consistent.





b) Y is distributed normally as n-%26gt; infinity, by the Central Limit Theorem. And, of course, it has a binomial distribution in general. The point they are going for with this one is that you can approximate the binomial distribution with the normal distribution for large N.





c) The distribution of Phat is normally distributed. Phat is the same as Y/n, where Y is the random variable from part (b).