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.
(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.