A continuous random variable X has distribution D on the real numbersR with distribution function12 arctan rFx (x) = P(X < x) =1+(a) Find the density function fx(x) = F'<(x) of this distribution.(b) Sketch this density function fx(x) and the density function of astandard Normal N(0, 1) distribution on the same graph.(c) Evaluate the mean integral / xfx(x) dx for X.-00(d) Find the Inverse-CDF function F' for this distribution.(e) Given the sample u of 10 independent Uni((0, 1)) random vari-ables, whereu = (0.23, 0.09, 0.63, 0.78, 0.40, 0.11, 0.5, 0.44, 0.86, 0.59),generate a sample of 10 independent simulations ¤1,..., T10 of D,and calculate the sample mean and sample variance.(f) If n independent samples x1, ... , Tn of D are generated in thisway, does the sample mean T = E *; tend to 0, the centre ofthe distribution D, as n → oo? Give reasons.

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Answered: A continuous random variable X has…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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