Thread: Can help with these statistical theory questions?

Can anyone shed some light on the below:


1. Consider a set with N distinct members, and a function f defined on Q that takes the values 0, 1 such that (1/N)*Sum(over xEQ) of f(x) = p. For a subset S of Q of size n, define the sample proportion


p := p(S) = (1/N)*Sum(over xES) of f(x)


If each subset of size n is chosen with equal probability, calculate the expectation and
standard deviation of the random variable p.


2.
(a) Let X-N(0, 1) be a normally distributed random variable with mean 0 and
variance 1. Suppose that x E R, x > 0. Find upper and lower bounds for the conditional expectation


E(X | X >x)


(b) Now suppose that X has a power law distribution, P(X >x) = a*(x)^-b, for x>x0>0, and some a> 0, b> 1. Calculate the conditional expectation
E(X|X>x), x >x0


Many thanks in advance.

The only thing I can think of for the first one is expectation of \hat{p} = p and the limit of the variance of \hat{p} as n -> N is 0. The variance of p is 0. Don't think this is correct though.