# How would I differentiate this? (Expectation)

Having a bit of trouble with expectations/probability. $$U = E_0 [u [C_0] + u [C_1]]$$ How do I differentiate this equation with respect to $C_0$? $E_0$ is the expectation, and $u$ is like the function of $C_0$ and $C_1$.

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Also asked on stats.SE – Dilip Sarwate Feb 2 '12 at 1:27
John, maybe you could expand a little on the conditions on $C_0$ and $C_1$ here? I am having trouble seeing why the question makes sense honestly. What is the measure of $E_0$? Is it the law of $C_0$? Are $C_0$ and $C_1$ real valued? – Chris Janjigian Feb 2 '12 at 2:05

Expectation on C0 is C0 because it is known at time 0. The utility function becomes U=u(C0) + E0 (u(C1)) dU/dC0 = u'(c0) + E0( u'(c1)*dc1/dc0)

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Unfortunately, I cannot comment on your question but I will update this attempt of an answer:

To be able to answer your question we need to clarify the following: With respect to which random variable are you taking the average? If the notation $E_0[\cdot]$ is the expectation with respect to the random variable $C_0$, then the quantity you mentioned above is independent of $C_0$.

Consider the following:

$$U= E_0 [u [C_0] + u [C_1]]\\ =\int dC_0\ \left(u[C_0]+u[C_1]\right)p(C_0)\\ =\int dC_0\ u(C_0)p(C_0)+u[C_1]\\ =\mathrm{a\ function\ of}\ C_1$$

I have assumed that $C_1$ is independent of $C_0$ in the above. The derivative of the expectation value wrt $C_0$ is thus zero.

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