Let x be a random variable with density function f and [a,b] be the possible set of values of x.

Let w(x) be a value associated with each value of x. I.e. w() is not known, and suppose that U(w(x)) is a \(\displaystyle C^{\infty} \) function.

Let \(\displaystyle H(w(x)) = \int_{a}^b f(x)U(w(x))dx \)

Find the derivative of H with respect to w over all x.

I don't know where to begin because I'm not even sure I understand the question.

I suspect the answer is f(x)U'(w(x)) only because it would be analogous to the case with finitely many possible value of x, but I dont know how to prove that.

Let w(x) be a value associated with each value of x. I.e. w() is not known, and suppose that U(w(x)) is a \(\displaystyle C^{\infty} \) function.

Let \(\displaystyle H(w(x)) = \int_{a}^b f(x)U(w(x))dx \)

Find the derivative of H with respect to w over all x.

I don't know where to begin because I'm not even sure I understand the question.

I suspect the answer is f(x)U'(w(x)) only because it would be analogous to the case with finitely many possible value of x, but I dont know how to prove that.

Last edited: