Suppose F: R→ [0, 1] is non-decreasing, right- continuous and such that limc→→∞ F(c) = 0 and limc→∞ F(c) = 1. Use the Caratheodory extension theorem to construct a probability measure P on (R,B) such that F is the cumulative distribution function of the random variable X: (R,B) → (R,B), X(x) = x when the domain is equipped with measure P.
Suppose F: R→ [0, 1] is non-decreasing, right- continuous and such that limc→→∞ F(c) = 0 and limc→∞ F(c) = 1. Use the Caratheodory extension theorem to construct a probability measure P on (R,B) such that F is the cumulative distribution function of the random variable X: (R,B) → (R,B), X(x) = x when the domain is equipped with measure P.
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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Question
M5
![Suppose F: R → [0, 1] is non-decreasing, right-
continuous and such that limc→-o F(c) = 0 and
limc→o F(c) = 1. Use the Caratheodory extension
theorem to construct a probability measure P on
(R,B) such that F is the cumulative distribution
function of the random variable X: (R,B) → (R,B),
X(x) = x when the domain is equipped with
%3D
measure P.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F516f1446-211a-46d9-a124-614103a7f612%2F002fe9bf-5f8e-4197-a425-d55c64af2be7%2Futjwf8f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Suppose F: R → [0, 1] is non-decreasing, right-
continuous and such that limc→-o F(c) = 0 and
limc→o F(c) = 1. Use the Caratheodory extension
theorem to construct a probability measure P on
(R,B) such that F is the cumulative distribution
function of the random variable X: (R,B) → (R,B),
X(x) = x when the domain is equipped with
%3D
measure P.
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