Q*: When do we have equality in Jensen's inequality? a) Assume the following a differentiable function...

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Q*: When do we have equality in Jensen's inequality? a) Assume the following a differentiable function f is a strictly concave up function on an interval [a, b]. That is, for each xo [a, b], f(x) ≥ ƒ(x0) + ƒ'(xo)(x − xo)holds for every x € [a, b] and the equality holds only when £ = 0. Now, for a finite set I, {Pi}ieI, {i}iЄI with 0<pi ≤ 1, Σiel Pi = When do we have Ciel Pif(ui) = Lielf(piui)? 1 and u₂ € [a, b]. b) Use the above consequence to show that for random variables X and Y taking finite many values, H(X+Y) = H(X) + H(Y) only when X and Y are independent. Q*: When do we have equality in Jensen's inequality? a) Assume the following a differentiable function f is a strictly concave up function on an interval [a, b]. That is, for each xo [a, b], f(x) ≥ ƒ(x0) + ƒ'(xo)(x − xo)holds for every x € [a, b] and the equality holds only when £ = 0. Now, for a finite set I, {Pi}ieI, {i}iЄI with 0<pi ≤ 1, Σiel Pi = When do we have Ciel Pif(ui) = Lielf(piui)? 1 and u₂ € [a, b]. b) Use the above consequence to show that for random variables X and Y taking finite many values, H(X+Y) = H(X) + H(Y) only when X and Y are independent.

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A a For equality to hold in Jensens inequality we must have that all of the uis are equal ie u1 ... View the full answer