2. Optimization: Consider the functional entropy defined as S[p]dxp(x)logp(x) where p(x) is a well defined probability density for xR+{0}. Suppose that all the knowledge we have about p(x) is summarized in the following properties: 1=dxp(x)=dxp(x)x where R is known. By using the definition of the functional derivative of a functional F (introduced in lecture 7): p(x0)F[p]ddF[p(x)+(xx0)]=0 where p is a suitable probability density, R is a real number and (xx0) is the Dirac's delta function centered at x0, demonstrate that the probability density that minimizes the functional entropy (1) subject to the constraints (2) and (3) is an exponential distribution: P(x)=1ex/. (30 marks) 2. Optimization: Consider the functional entropy defined as S[p]dxp(x)logp(x) where p(x) is a well defined probability density for xR+{0}. Suppose that all the knowledge we have about p(x) is summarized in the following properties: 1=dxp(x)=dxp(x)x where R is known. By using the definition of the functional derivative of a functional F (introduced in lecture 7): p(x0)F[p]ddF[p(x)+(xx0)]=0 where p is a suitable probability density, R is a real number and (xx0) is the Dirac's delta function centered at x0, demonstrate that the probability density that minimizes the functional entropy (1) subject to the constraints (2) and (3) is an exponential distribution: P(x)=1ex/. (30 marks)