For a probability density function p(x) >0 on the interval (-∞, ∞) the entropy functional S[p] is given by S[p] = - fo dx p(x) log p(x). Suppose that the function p(x) is subject to the constraints 8 T dx f(x) p(x) = c, ∞ p dx p(x) = 1 and where f(x) is a fixed function and c is a constant. Show that is a stationary path for S[p] is given by p(x) = exp(-1 - \ - μf(x)), where A and μ are Lagrange multipliers.
For a probability density function p(x) >0 on the interval (-∞, ∞) the entropy functional S[p] is given by S[p] = - fo dx p(x) log p(x). Suppose that the function p(x) is subject to the constraints 8 T dx f(x) p(x) = c, ∞ p dx p(x) = 1 and where f(x) is a fixed function and c is a constant. Show that is a stationary path for S[p] is given by p(x) = exp(-1 - \ - μf(x)), where A and μ are Lagrange multipliers.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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![For a probability density function p(x) > 0 on the interval
(-∞, ∞) the entropy functional S[p] is given by
-I dx p(x) log p(x).
S[p]
=
Suppose that the function p(x) is subject to the constraints
Lo de
[* dx p(x) 1 and
=
-∞
dx f(x)p(x) = c,
where f(x) is a fixed function and c is a constant.
Show that is a stationary path for S[p] is given by
p(x) = exp(-1-\- µf(x)),
where λ and u are Lagrange multipliers.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F59d44c96-efb1-4f3c-83b3-5a6a84cf94cb%2F9b21dce9-7afb-4262-820d-c3aa6682e0e1%2Flnpwxin_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
-I dx p(x) log p(x).
S[p]
=
Suppose that the function p(x) is subject to the constraints
Lo de
[* dx p(x) 1 and
=
-∞
dx f(x)p(x) = c,
where f(x) is a fixed function and c is a constant.
Show that is a stationary path for S[p] is given by
p(x) = exp(-1-\- µf(x)),
where λ and u are Lagrange multipliers.
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