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Probability density function

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#4. The probability density function of X, the lifetime of a certain type of electronic device (measured in hours), is given by

f (x) = 10/x2 x > 10
f (x) = 0 x ≤ 10

(a) Find P{X > 20}.
(b) What is cumulative distribution function of X?
.. (see attachment)

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this shows how to find the cumulative distribution function.

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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.

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