re dealing with an improper integral. We will now see that the expectation of a Cauchy random ariable is undefined by showing the following: a) Show that G(a,b) := مر _ydy = = = 1 (2+1). In 7 (141²) + b) Prove that the limit of G(a, b) to the real line (i.e., the multivariable limit (a, b) → (-∞, ∞0)) does not exist by showing that we can find multiple values for the limit. Conclude that the expectation of a Cauchy random variable is undefined
re dealing with an improper integral. We will now see that the expectation of a Cauchy random ariable is undefined by showing the following: a) Show that G(a,b) := مر _ydy = = = 1 (2+1). In 7 (141²) + b) Prove that the limit of G(a, b) to the real line (i.e., the multivariable limit (a, b) → (-∞, ∞0)) does not exist by showing that we can find multiple values for the limit. Conclude that the expectation of a Cauchy random variable is undefined
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Problem 4 please
Transcribed Image Text:Problem 3 (p. 310 #6). Cauchy distribution. Suppose that a particle is fired from the origin
in the (x, y) plane in a straight line at a random angle where is chosen uniformly from the
interval (-) Show that the random variable Y = y-coordinate of the intersection between the
point and the vertical line z = 1 has density
fy (y)
1
π(1 + y²)*
This is called the Cauchy distribution.
Problem 4. When attempting to compute the expectation of the Cauchy random variable in the
previous problem, there are some subtleties that must be overcome, or at least understood since we
are dealing with an improper integral. We will now see that the expectation of a Cauchy random
variable is undefined by showing the following:
a) Show that
y dy
G(a,b) := "7 (147²)
= = = ( 2² + 1).
In
2π
a²+
b) Prove that the limit of G(a, b) to the real line (i.e., the multivariable limit (a, b) → (-∞0, ∞))
does not exist by showing that we can find multiple values for the limit. Conclude that the
expectation of a Cauchy random variable is undefined.
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