Describe the probability density function

Question

Discuss the below:

If X1,X2,..., Xn, are (iid) , from a distribution with mean μ and variance σ². Define the sample mean as

Xbar = (X1+X2+...+Xn) / n

(a) Show that the mean and variances of the probability density function of Xbar are given as E(X‾) = μ

Var(X‾) = (σ²)/n

b) What is the central limit theorem?

c) If n, is large, can you describe fully, the probability density function of Xbar?

d) Can you describe fully the probability density function of the variable y = e^X‾ This random variable is called a lognormal random variable, and is used very frequently in finance.