# 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.

https://brainmass.com/statistics/probability/statistics-probability-density-function-central-limit-theorem-296948

#### Solution Preview

Please see the attached file.

Given are iid random variables with mean and variance 2. This means and ;

a) Mean of is given by

That is

Variance of is given by

That is

NB. Note that

b) Central Limit Theorem (Lindberg Levy form): If are independent and ...

#### Solution Summary

The expert examines the probability density function for central limit theorem.