See the attached file.
If X1,X2,..., Xn, are (iid) , from a distribution with mean μ and variance σ^2. 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(Xbar) = μ
Var(Xbar) = (σ^2)/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^Xbar? This random variable is called a lognormal random variable, and is used very frequently in finance.
Please see the attached file.
Given are iid random variables with mean and variance 2. This means and ;
a) Mean of is given by
Variance of is given by
NB. Note that
b) Central Limit Theorem (Lindberg Levy form): If are independent and ...
The expert examines the probability density function for central limit theorem.