4. Consider a probability distribution p(x) defined on the whole real line. Suppose that we know only the first three power moments of p: μn = = drz"p(x) for n 0, 1, 2. a) Write down the entropy functional with three Lagrange multipliers to handle the constraints (moment conditions). b) Find the maximum entropy solution to part a) using the Euler-Lagrange equation
4. Consider a probability distribution p(x) defined on the whole real line. Suppose that we know only the first three power moments of p: μn = = drz"p(x) for n 0, 1, 2. a) Write down the entropy functional with three Lagrange multipliers to handle the constraints (moment conditions). b) Find the maximum entropy solution to part a) using the Euler-Lagrange equation
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Transcribed Image Text:4. Consider a probability distribution p(x) defined on the whole real line. Suppose that
we know only the first three power moments of p:
for n = 0, 1, 2.
μn =
=* dxx" p(x)
a) Write down the entropy functional with three Lagrange multipliers to handle the
constraints (moment conditions).
b) Find the maximum entropy solution to part a) using the Euler-Lagrange equation
c) Give an analytic expression for the maximum-entropy density Pм(x) in terms of the
known moments.
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