Assume that X is a random variable with a probability density function
Find the constant c;
Statistics: Probability density function, central limit theorem
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.