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Suppose $x$ has a binomial distribution with chance $\alpha$ drawn $k$ times, and let $f(x)$ be a positive convex real valued function. I would like to evaluate

$$\frac{\partial}{\partial \alpha} \mathbb{E} [f(x)]$$

Ultimately, this is for the following purpose: Suppose $\alpha^*(y)$ is defined by the equation

$$\mathbb{E} [f(x) \mid \alpha] \equiv \alpha + y$$

I am interested in studying the implicit function $\alpha^*(y)$ so defined. Any tips to characterize it would be greatly appreciated.

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  • $\begingroup$ What does the notation $\mathbb{E}[ \cdot \mid \alpha]$ mean? Please tell us more about what this problem is for, but the paper referenced at mathoverflow.net/questions/21858/… by Sah ("The Effects of Child Mortality Changes on Fertility Choice and Parental Welfare", equations 4 - 7) offers a property which sufficed for a similar problem I was tackling. Namely, \begin{align*} \frac{\partial}{\partial \alpha} \mathbb{E}[f(x)] = k \cdot \sum_{j=0}^{k-1} {k-1 \choose j} \alpha^j (1-\alpha)^{k-1-j} (f(x+1)-f(x)). \end{align*} (cont.) $\endgroup$ – user39430 Sep 4 '13 at 17:08
  • $\begingroup$ Lots of other nice properties are derived in this paper, and the answers to that question offer some further reading. Edit: See also mathoverflow.net/questions/122897/… which has much more analysis of this type. $\endgroup$ – user39430 Sep 4 '13 at 17:08
  • $\begingroup$ The notation $\mathbb E[f(x)\mid\alpha]$ is a way of writing the expected value in which its dependence on $\alpha$ is given a place in the notation, so that, for example, one can write expressions like $\mathbb E[f(x)\mid\alpha]/\mathbb E[f(x)\mid\alpha_0]$. Suppose that 20 pages after you first write about the random variable $x$ whose distribution depends on $\alpha$, it becomes appropriate to assign a probability distribution to $\alpha$. Then the notation refers to the conditional expected value while remaining consistent with the way it was used earlier. $\endgroup$ – Michael Hardy Oct 4 '13 at 21:16
  • $\begingroup$ +1. I changed \;|\; to \mid. I think that is standard. $\endgroup$ – Michael Hardy Oct 4 '13 at 21:19
  • $\begingroup$ So $\alpha^*(y)$ is the value of $\alpha$ for which the bias of $f(x)$ as an estimator of $\alpha$ is $y$. $\endgroup$ – Michael Hardy Oct 4 '13 at 21:20

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