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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 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.

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    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.