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# Statistics: Probability density function, central limit theorem

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If X1,X2,..., Xn, are (iid) , from a distribution with mean &#956; and variance &#963;^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) = &#956;
Var(Xbar) = (&#963;^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.

##### Solution Summary

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

##### Solution Preview

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 ...

##### Measures of Central Tendency

This quiz evaluates the students understanding of the measures of central tendency seen in statistics. This quiz is specifically designed to incorporate the measures of central tendency as they relate to psychological research.