A continuous random variable X has distribution D on the real numbers R with distribution function Fx(x) = P(X < x) = 1 2 arctan r 1+ (a) Find the density function fx(x) = Fx(x) of this distribution. (b) Sketch this density function fx(x) and the density function of a standard Normal N(0, 1) distribution on the same graph. (c) Evaluate the mean integral / afx(x) dx for X. (d) Find the Inverse-CDF function Fx' for this distribution. (e) Given the sample u of 10 independent Uni((0, 1)) random vari- ables, where u = (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,... , ¤10 of D, and calculate the sample mean and sample variance. (f) If n independent samples r1, ..., xn of D are generated in this way, does the sample mean 7 = :E,"; tend to 0, the centre of the distribution D, as n →? Give reasons.
A continuous random variable X has distribution D on the real numbers R with distribution function Fx(x) = P(X < x) = 1 2 arctan r 1+ (a) Find the density function fx(x) = Fx(x) of this distribution. (b) Sketch this density function fx(x) and the density function of a standard Normal N(0, 1) distribution on the same graph. (c) Evaluate the mean integral / afx(x) dx for X. (d) Find the Inverse-CDF function Fx' for this distribution. (e) Given the sample u of 10 independent Uni((0, 1)) random vari- ables, where u = (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,... , ¤10 of D, and calculate the sample mean and sample variance. (f) If n independent samples r1, ..., xn of D are generated in this way, does the sample mean 7 = :E,"; tend to 0, the centre of the distribution D, as n →? Give reasons.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:A continuous random variable X has distribution D on the real numbers
R with distribution function
1
2 arctan r
Fx (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 a
standard 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, where
u = (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 this
way, does the sample mean T = E *; tend to 0, the centre of
the distribution D, as n → oo? Give reasons.
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