Show further that the Cauchy distribution with p(x) = 1 1 π 1 + x² ?? is a solution for the case f(x) = ln(1 + x²) and c = 2ln 2, and obtain the Lagrange multipliers in this case. You may assume that the methods of the calculus of variations apply on the infinite interval (–∞, ∞). The values of the following integrals may be useful: ୮ x dx 1 1+x² = π, مر In(1 + x²) dx = 2π In 2. 1+x² For a probability density function p(x) > 0 on the interval (-∞, ∞) the entropy functional S[p] is given by S[p] = − (** dx p(x) logp(x). - L Suppose that the function p(x) is subject to the constraints Lo dr p(x) dx p(x) = 1 and L dx f(x)p(x) = c, -00 -∞ where f(x) is a fixed function and c is a constant. stationary path for S[p] is given by - p(x) = exp(−1 — λ — µƒ(x)), where and μ are Lagrange multipliers.
Show further that the Cauchy distribution with p(x) = 1 1 π 1 + x² ?? is a solution for the case f(x) = ln(1 + x²) and c = 2ln 2, and obtain the Lagrange multipliers in this case. You may assume that the methods of the calculus of variations apply on the infinite interval (–∞, ∞). The values of the following integrals may be useful: ୮ x dx 1 1+x² = π, مر In(1 + x²) dx = 2π In 2. 1+x² For a probability density function p(x) > 0 on the interval (-∞, ∞) the entropy functional S[p] is given by S[p] = − (** dx p(x) logp(x). - L Suppose that the function p(x) is subject to the constraints Lo dr p(x) dx p(x) = 1 and L dx f(x)p(x) = c, -00 -∞ where f(x) is a fixed function and c is a constant. stationary path for S[p] is given by - p(x) = exp(−1 — λ — µƒ(x)), where and μ are Lagrange multipliers.
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:Show further that the Cauchy distribution with
p(x)
=
1 1
π 1 + x²
??
is a solution for the case f(x) = ln(1 + x²) and c = 2ln 2, and
obtain the Lagrange multipliers in this case.
You may assume that the methods of the calculus of variations
apply on the infinite interval (–∞, ∞). The values of the following
integrals may be useful:
୮
x
dx
1
1+x²
= π,
مر
In(1 + x²)
dx
= 2π In 2.
1+x²
![For a probability density function p(x) > 0 on the interval
(-∞, ∞) the entropy functional S[p] is given by
S[p] = − (** dx p(x) logp(x).
-
L
Suppose that the function p(x) is subject to the constraints
Lo dr p(x)
dx p(x) = 1 and
L
dx f(x)p(x) = c,
-00
-∞
where f(x) is a fixed function and c is a constant.
stationary path for S[p] is given by
-
p(x) = exp(−1 — λ — µƒ(x)),
where and μ are Lagrange multipliers.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa6c8ed7d-75cc-4e27-869e-3ad6a1efc0b4%2F6fabb833-a978-4324-9499-46cf02444f0d%2F9qz8ilc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:For a probability density function p(x) > 0 on the interval
(-∞, ∞) the entropy functional S[p] is given by
S[p] = − (** dx p(x) logp(x).
-
L
Suppose that the function p(x) is subject to the constraints
Lo dr p(x)
dx p(x) = 1 and
L
dx f(x)p(x) = c,
-00
-∞
where f(x) is a fixed function and c is a constant.
stationary path for S[p] is given by
-
p(x) = exp(−1 — λ — µƒ(x)),
where and μ are Lagrange multipliers.
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